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Ty^ojU.
A
HARVARD COLLEGE
LIBRARY
FROM THE
FARRAR FUND
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MMawry <^ibr AtMhuwI, /oiM Farrar,
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A TEEATISE
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UNIVERSAL ALGEBEA.
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lonion: c J. CLAY and sons,
CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.
•Iwgoto: 268, AB6TLB STREET.
E(tp>ig: F. A. BROCKHAUS.
000 IBorfc: THE MACMILLAN OOMPANT.
ISombaR: E. SEYMOUR HALE.
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A TEEATISE
ON
UNIVEKSAL ALGEBEA
WITH APPLICATIONS.
BY
ALFRED NORTH WHITEHEAD, M.A.
FELLOW AND LECTURER OF TRINITY COLLKOB, CAKBRIDOE.
VOLUME I.
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CAMBRIDGE:
AT THE UNIVERSITY PRESS.
1898
[All RighU reierved.]
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PEEFACE.
TT is the purpose of this work to present a thorough investigation of the
various systems of Symbolic Reasoning allied to ordinary Algebra. The
chief examples of such systems are Hamilton's Quaternions, Grassmann's
Calculus of Extension^ and Boole's Symbolic Logic. Such algebras have
an intrinsic value for separate detailed study ; also they are worthy of a
comparative study, for the sake of the light thereby thrown on the general
theory of symbolic reasoning, and on algebraic symbolism in particular.
The comparative study necessarily presupposes some previous separate
study, comparison being impossible without knowledge. Accordingly after
the general principles of the whole subject have been discussed in Book I.
of this volume, the remaining books of the volume are devoted to the separate
study of the Algebra of Symbolic Logic, and of Grassmann's Calculus of
Extension, and of the ideas involved in them. The idea of a generalized
conception of space has been made prominent, in the belief that the
properties and operations involved in it can be made to form a uniform
method of interpretation of the various algebras.
Thus it is hoped in this work to exhibit the algebras both as systems
of symbolism, and also as engines for the investigation of the possibilities
of thought and reasoning connected with the abstract general idea of space.
A natural mode of comparison. between the algebras is thus at once provided
by the unity of the subject-matters of their interpretation. The detailed
comparison of their symbolic structures has been adjourned to the second
volume, in which it is intended to deal with Quaternions, Matrices, and the
general theory of Linear Algebras. This comparative anatomy of the subject
was originated by B. Peirce's paper on Linear Associative Algebra*, and has
been carried forward by more recent investigations in Germany.
* Firat read before the National Academy of Soienoes in Washington, 1871, and repabliahed
in the American Journal of Mathematics, vol. iv., 1881.
i
VI PREFACE.
The general name to be given to the subject has caused me much thought :
that finally adopted, Universal Algebra, has been used somewhat in this
signification by Sylvester in a paper, Lectures on the Principles of Universal
Algebra, published in the American Journal of Mathematics, vol. vi., 1884.
This paper however, apart from the suggestiveness of its title, deals ex-
plicitly only with matrices.
Universal Algebra has been looked on with some suspicion by many
mathematicians, as being without intrinsic mathematical interest and as
being comparatively useless as an engine of investigation. Indeed in this
respect Symbolic Logic has been peculiarly unfortunate; for it has been
disowned by many logicians on the plea that its interest is mathematical, and
by many mathematicians on the plea that its interest is logical. Into the
quarrels of logicians I shall not be rash enough to enter. Also the nature of
the interest which any individual mathematician may feel in some branch of
his subject is not a matter capable of abstract argumentation. But it may
be shown, I think, that Universal Algebra has the same claim to be a serious
subject of mathematical study as any other branch of mathematics. In order
to substantiate this claim for the importance of Universal Algebra, it is
necessary to dwell shortly upon the fundamental nature of Mathematics.
Mathematics in its widest signification is the development of all types of
formal, necessary, deductive reasoning.
The reasoning is formal in the sense that the meaning of propositions
forms no part of the investigation. The sole concern of mathematics is the
inference of proposition from proposition. The justification of the rules of
inference in any branch of mathematics is not properly part of mathematics :
* it is the business of experience or of philosophy. The business of
mathematics is simply to follow the rule. In this sense all mathematical
reasoning is necessary, namely, it has followed the rule.
Mathematical reasoning is deductive in the sense that it is based upon
definitions which, as far as the validity of the reasoning is concerned (apart
> from any existential import), need only the test of self-consistency. Thus no
external verification of definitions is required in mathematics, as long as it is
considered merely as mathematics. The subject-matter is not necessarily
first presented to the mind by definitions : but no idea, which has not been
completely defined as far as concerns its relations to other ideas involved in
the subject-matter, can be admitted into the reasoning. Mathematical
definitions are always to be construed as limitations as well as definitions;
/
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PREFACE. VU
namely, the properties of the thing defined are to be considered for the
purposes of the argument as being merely those involved in the definitions.
Mathematical definitions either possess an existential import or are
conventional. A mathematical definition with an existential import is the
result of an act of pure abstraction. Such definitions are the starting points
of applied mathematical sciences; and in so far as they are given this
existential import, they require for verification more than the mere test
of self-consistency.
Hence a branch of applied mathematics, in so far as it is applied, is not
merely deductive, Unless in some sense the definitions are held to be
guaranteed a priori as being true in addition to being self-consistent.
A conventional mathematical definition has no existential import. It sets
before the mind by an act of imagination a set of things with fully defined
self-consistent types of relation. In order that a mathematical science of any
importance may be founded upon conventional definitions, the entities created
by them must have properties which bear some afiinity to the properties
of existing things. Thus the distinction between a mathematical definition
with ibi existential import and a conventional definition is not always very
obvious fi*om the form in which they are stated. Though it is possible
to make a definition in form unmistakably either conventional or existential,
there is often no gain in so doing. In such a case the definitions and resulting
propositions can be construed either as refeiiing to a world of ideas created
by convention, or as referring exactly or approximately to the world of existing
things. The existential import of a mathematical definition attaches to it, if
at all, qu& mixed mathematics ; qu& pure mathematics, mathematical defi-
nitions must be conventional*.
Historically, mathematics has, till recently, been confined to the theories
of Number, of Quantity (strictly so-called), and of the Space of common
experience. The limitation was practically justified : for no other large
systems of deductive reasoning were in existence, which satisfied our
definition of mathematica The introduction of the complex quantity of
ordinary algebra, an entity which is evidently based upon conventional
definitions, gave rise to the wider mathematical science of to-day. The
realization of wider conceptions has been retarded by the habit of mathe-
maticians, eminently useful and indeed necessary for its own purposes, of
extending ail names to apply to new ideas as they arise. Thus the name
* Cf. Or&ssmann, Ausdehnungslehre von 1S44, Einleitang.
1
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viii PREFACE.
of quantity was transferred from the quantity, strictly so called, to the
generalized entity of ordinary algebra, created by conventional definition,
which only includes quantity (in the strict sense) as a special case.
Ordinary algebra in its modem developments is studied as being a large "
body of propositions, inter-related by deductive reasoning, and based upon 'i
conventional definitions which are generalizations of fiindamental conceptions. |
Thus a science is gradually being created, which by reason of its fundamental
character has relation to almost every event, phenomenal or intellectual,
which can occur. But these reasons for the study of ordinary Algebra apply i
to the study of Universal Algebra ; provided that the newly invented '
algebras can be shown either to exemplify in their sjrmbolism, or to represent
in their interpretation interesting generalizations of important systems of i
ideas, and to be useful engines of investigation. Such algebras are j j
mathematical sciences, which are not essentially concerned with number
or quantity ; and this bold extension beyond the traditional domain of pure
quantity forms their peculiar interest. The ideal of mathematics should be
to erect a calculus to facilitate reasoning in connection with every province of
thought, or of external experience, in which the succession of thoughts, or of
events can be definitely ascertained and precisely stated. So that all serious
thought which is not philosophy, or inductive re&soning, or imaginative
literature, shall be mathematics developed by means of a calculus.
It is the object of the present work to exhibit the new algebras, in their
detail, as being usefiil engines for the deduction of propositions ; and in their
several subordination to dominant ideas, as being representative symbolisms
of fundamental conceptions. In conformity with this latter object I have
not hesitated to compress, or even to omit, developments and applications
which are not allied to the dominant interpretation of any algebra. Thus
unity of idea, rather than completeness, is the ideal of this book. I am
convinced that the comparative neglect of this subject during the last forty
years is partially due to the lack of unity of idea in its presentation.
The neglect of the subject is also, I think, partially due to another defect
in its presentation, which (for the want of a better word) I will call the lack
of independence with which it has been conceived. I will proceed to explain
my meaning.
Every method of research creates its own applications : thus Analytical
Geometry is a different science from Synthetic Geometry, and both these
sciences are diflTerent from modem Projective Geometry. Many propositions
\
PREFACE. IX
are identical in all three sciences, and the general subject-matter, Space, is
the same throughout. But it would be a serious mistake in the development
of one of the three merely to take a list of the propositions as they occur in
the others, and to endeavour to prove them by the methods of the one in
hand. Some propositions could only be proved with great difficulty, some
could hardly even be stated in the technical language, or symbolism, of the
special branch. The same applies to the applications of the algebras in this
book. Thus Grassmann's Algebra, the Calculus of Extension, is applied to
Descriptive Geometry, Line Geometry, and Metrical Geometry, both non-
Euclidean and Euclidean. But these sciences, as here developed, are not
the same sciences as developed by other methods, though they apply to the
same general subject-matter. Their combination here forms one new and
distinct science, as distinct from the other sciences, whose general subject-
matters they deal with, as is Analytical Geometry from Pure Geometry.
This distinction, or independence, of the application of any new algebra
appears to me to have been insufficiently realized, with the result that the
developments of the new Algebras have been cramped.
In the use of symbolism I have endeavoured to be very conservative.
Strange symbols are apt to be rather an encumbrance than an aid to
thought: accordingly I have not ventured to disturb any well-established
notation. On the other hand I have not hesitated to introduce fresh symbols
when they were required in order to express new ideas.
This volume is divided into seven books. In Book I. the general prin-
ciples of the whole subject are considered. Book II. is devoted to the
Algebra of Symbolic Logic; the results of this book are not required in any
of the succeeding books of this volume. Book III. is devoted to the general
principles of addition and to the theory of a Positional manifold, which is a
generalized conception of Space of any number of dimensions without the
introduction of the idea of distance. The comprehension of this book is
essential in reading the succeeding books. Book IV. is devoted to the
principles of the Calculus of Extension. Book V. applies the Calculus of
Extension to the theory of forces in a Positional manifold of three dimensions.
Book VI. applies the Calculus of Extension to Non-Euclidean Geometry,
considered, after Cayley, as being the most general theory of distance in a
Positional manifold; the comprehension of this book is not necessary in
reading the succeeding book. Book VII. applies the Calculus of Extension
to ordinary Euclidean Space of three dimensions.
\
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X PREFACE.
It would have been impossible within reasonable limits of time to have
made an exhaustive study of the many subjects, logical and mathematical,
on which this volume touches ; and, though the writing of this volume has
been continued amidst other avocations since the year 1890, I cannot
pretend to have done so. In the subject of pure Logic I am chiefly indebted
to Mill, Jevons, Lotze, and Bradley; and in regard to Symbolic Logic to
Boole, Schroder and Venn. Also I have not been able in the footnotes to
this volume adequately to recognize my obligations to De Morgan's writings,
both logical and mathematical. The subject-matter of this volume is not
concerned with Quaternions; accordingly it is the more necessary to mention
in this preface that Hamilton must be regarded as a founder of the
science of Universal Algebra. He and De Morgan (c£ note, p. 131)
were the first to express quite clearly the general possibilities of algebraic |
symbolism. j
The greatness of my obligations in this volume to Grassiliaann will be '
understood by those who have mastered his two AusdehnungslehrSs^ The
technical development of the subject is inspired chiefly by his work of 1862,
but the underlying ideas follow the work of 1844. At the same time I have
tried to extend his Calculus of Extension both in its technique and in its
ideas. But this work does not profess to be a complete interpretation of
Grassmann's investigations, and there is much valuable matter in his
Ausdehnungslehres ivhich it has not fallen within my province to touch
upon. Other obligations, as far as I am aware of them, are mentioned as
they occur. But the book is the product of a long preparatory period- of
thought and miscellaneous reading, and it was only gradually that the
subject in its full extent . shaped itself in my mind ; since then the various
parts of this volume have been systematically deduced* according to the
methods appropriate to them here, with hardly any aid from other works.
This procedure was necessary, if any unity of idea was to be preserved, owing
to the bewildering variety of methods and points of view adopted by writers
on the various subjects of this volume. Accordingly there is a possibility of
some oversights, which I should very much regret, in the attribution of ideas
and methods to their sources. I should like in this connection to mention
the names of Arthur Buchheim and of Homersham Cox as the mathematicians
whose writings have chiefly aided me in the development of the Calculus of
Extension (cf. notes, pp. 248, 346, 370, and 575). In the development of
Non-Euclidean Geometry the ideas of Cayley, Klein, and Clifford have been
r '"«
PREFACE. XI
chiefly followed ; and in the development of the theory of Systems of Forces
I am indebted to Sir R. S. Ball, and to Lindemann.
I have added unsystematieally notes to a few theorems or methods,
stating that they are, as far as I know, now enunciated for the first time.
These notes are unsystematic in the double sense that I have not made
a systematic search in the large literatures of the many branches of
mathematics with which this book has to do, and that I have not added
notes to every theorem or method which happens to be new to me.
My warmest thanks for their aid in the final revision of this volume are
due to Mr Arthur Berry, Fellow of King's College, to Mr W. E. Johnson,
of Eling's College, and Lecturer to the University in Moral Science, to
Prof Fors3rth, Sadlerian Professor to the University, who read the first three
books in manuscript, and to the Hon. B, Russell, Fellow of Trinity College,
who has read many of the proofs, especially in the parts connected with
^o Non-Euclidean Geometry.
Mr Johnson not only read the proofs of the first three books, and made
many important suggestions and corrections, but also generously placed at
my disposal some work of his own on Symbolic Logic, which will be found
duly incorporated with acknowledgements.
Mr Berry throughout the printing of this volume has spared himself no
trouble in aiding me with criticisms and suggestions. He undertook the
extremely laborious task of correcting all the proofs in detail. Every page
has been improved either substantially or in expression owing to his
suggestions I cannot express too strongly my obligations to him both for
his general and detailed criticism.
The high efficiency of the University Press in all that concerns mathe-
matical printing, and the courtesy which I have received from its Officials,
also deserve grateful acknowledgements.
Gambbidob,
December, 1S97.
I
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CONTENTS.
The following Books and Chapters are not essential far the comprehension
of the subsequent paHs of this volume : Book II, Chapter V of Book IV,
Book VI,
BOOK I.
PRINCIPLES OF ALGEBRAIC SYMBOLISM.
CHAPTER I.
On the Nature of a Calculus.
ART. PA0E8
1. Signs 3—4
2. Definition of a Calculus 4 — 5
3. Equivalence 6 — 7
4. Operations 7 — 8
5. Substitutive Schemes 8 — 9
6. Conventional Schemes 9 — 10.
7. Uninterpretable Forms 10 — 12
CHAPTER 11.
Manifolds.
8. Manifolds 13—14
9. Secondary Properties of Elements 14-15
10. Definitions 15
11. Special Manifolds 16-17
52
xiv CONTENTS.
CHAPTER III.
Principles of Universal Alqebra.
ART. I*AOB8
12. Introductory 18
13. Equivalence 18—19
14. Principles of Addition 19 — 21
15. Addition 21—22
16. Principles of Subtraction 22—24
17. The Null Element 24—26
18. Steps 26
19. Multiplication 25—27
20. Orders of Algebraic Manifolds 27—28
21. The Nidi Element 28—29
22. Classification of Special Algebras 29 — 32
Note 32
BOOK 11.
THE ALGEBRA OF SYMBOLIC LOGIC.
CHAPTER I.
The Algebra of Symbolic Logic.
23. Formal Laws 35 — 37
24. Reciprocity between Addition and Multiplication .... 37 — 38
25. Interpretation 38—39
26. Elementary Propositions 39 — 41
27. Classification 41-^2
28. Incident Regions 42 — 44
CHAPTER IL
The Algebra of Symbolic Logic {c<mixnv/ed),
29. Development 46 — 47
30. Elimination 47—56
31. Solution of Equations with One Unknown 56 — 59
32. On Limiting and Unlimiting Equations 59 — 60
33. On the Fields of Expressions 60—65
34. Solution of Equations with More than One Unknown .... 65 — 67
36. Symmetrical Solution of Equations with Two Unknowns 67 — 73
36. Johnson's Method 73—76
37. Symmetrical Solution of Equations with Three Unknowns . 75 — 80
38. Subtraction and Division 80—82
CONTENTS. XV
CHAPTER m.
Existential Expressions.
ABT. PAGES
39. Existential Expressions 83 — 86
40. Umbral Letters 86-89
41. Elimination 89—91
42. Solutions of Existential Expressions with One Unknown . 91 — 92
43. Existential Expressions with Two Unknowns 93 — 94
44. Equations and Existential Expressions with One Unknown. . 94—96
46. Boole's General Problem 96-97
46. Equations and Existential Propositions with Many Unknowns 97-— 98
NoU 98
CHAPTER IV.
Application to Logic.
47. Propositions 99—100
48. Exclusion of Nugatory Forms 100—101
49. Syllogism 101—103
50. Symbolic Equivalents of Syllogisms 103 — 105
61. Generalization of Logic 106 — 106
CHAPTER V.
Propositional Interpretation.
52. Propositional Interpretation 107 — 108
53. Equivalent Propositions 108
54. Symbolic Representation of Complexes 108
55. Identification with the Algebra of Symbolic L<^c .... 108 — 111
56. Existential Expressions Ill
57. Symbolism of the Traditional Propositions Ill — 112
58. Primitive Predication 112—113
59. Existential Symbols and Primitive Predication 113—114
60. Propositions 114—115
Historical Note 115—116
XVI CONTENTS.
BOOK III.
POSITIONAL MANIFOLDS.
CHAPITER I.
Fundamental Propositions.
ABT. PAGES
61. Introductory 119
62. Intensity 119—121
63. Things repi-esenting Difierent Elements 121 — 122
64. Fundamental Propositions 122—125
66. Subregions 126 — 128
66. Loci 128—130
67. Surface Loci and Curve Loci 130—131
Note 131
CHAPTER II.
Straight Lines and Planes.
68. Introductory 132
69. Anharmonic Ratio 132
70. Homographic Ranges 133
71. Linear Transformations 133 — 136
72. Elementary Properties 136—137
73. Reference-Figiures 138 -139
74. Perspective 139—142
76. Quadrangles 142—143
CHAPTER III.
QUADRICS.
76. Introductory 114
77. Elementary Properties 144 — 146
78. Poles and Polars «... 146—147
79. Generating R^ons 147 — 148
80. Conjugate Coordinates 148 — 151
81. Quadriquadric Curve Loci 151 — 153
82. Closed Quadrics 163-165
83. Conical Quadric Surfaces 156—157
84. Reciprocal Equations and Conical quadrics 167 — 161
Note 161
CONTENTS. XVU
CHAPTER IV.
Intensity.
ABT. PA0B8
85. Defining Equation of Intensity 162—163
86. Locus of Zero Intensity 163—164
87. Plane Locus of Zero Intensity 164—166
88. Quadric Locus of Zero Intensity 166
89. Antipodal Elements and Opposite Intensities 166 — 167
90. The Intercept between Two Elements 167—168
Note 168
BOOK IV.
CALCULUS OF EXTENSION.
CHAPTER I.
Combinatorial Multiplicahon.
91. Introductory 171—172
92. Invariant Equations of Condition 172 — 173
93. Principles of Combinatorial Multiplication 173 — 17ft
94. Derived Manifolds 175—176
95. Extensive Magnitudes 176—177
96. Simple and Compoimd Extensive Magnitudes 177 — 178
97. Fundamental Propositions 178 — 180
Note 180
CHAPTER II.
Regressive Multiplication.
98. Progressive and Regressive Multiplication 181
99. Supplements 181—183
100. Definition of Regressive Multiplication . 183—184
lOL Pure and Mixed Products 184-185
102. Rule of the Middle Factor 185—188
103. Extended Rule of the Middle Factor 188—190
104. Regressive Multiplication independent of Reference-Elements . . 190—191
105. Proposition 191
106. Mtiller's Theorems 192—195
107. Applications and Examples 195—198
Note 198
xviii CONTENTS.
CHAPTER III.
Supplements.
ABT. PAGES
108. Supplementary R^ons 199
109. Normal Systems of Points 199—200
110. Extension of the Definition of Supplements 201—202
111. Different kinds of Supplements 202
112. Normal Points and Straight Lines 202—203
113. Mutually normal Regions 203—204
114. Self-normal Elements 204—206
115. Self-normal Planes 206
116. Complete Region of Three Dimensions 206 — 207
117. Inner Multiplication 207
118. Elementary Transformations 208
119. Rule of the Middle Factor 208
120. Important Formula 208—209
121. Inner Multiplication of Normal Regions 209
122. General Formula for Inner Multiplication 209—210
123. Quadrics 210—212
124. Plane-Equation of a Quadric 212—213
CHAPTER IV.
Descriptive Geometry.
125. Application to Descriptive Geometry 214
126. Explanation of Procedure 214—215
127. Illustration of Method 215
128. von Staudt's Construction 215 — 219
129. Grassmann's Constructions 219 — 223
130. Projection 224—228
CHAPTER V.
Descriptive Geometry of Conics and CuBica
131. General Equation of a Conic 229—231
132. Further Transformations 231—233
133. Linear Construction of Cubics 233
134. First Type of Linear Construction of the Cubic 233—236
135. Linear Construction of Cubic through Nine arbitrary Points . . 236—237
136. Second Type of Linear Construction of the Cubic .... 238—239
137. Third Type of Linear Construction of the Cubic 239—244
138. Fourth Type of Linear Construction of the Cubic .... 244—246
139. ChasW Construction 246—247
1
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CONTENTS.
XIX
CHAPTER VI.
Matrices.
ABT. PAOB8
140. Introductory 248
141. Definition of a Matrix 248—249
142. Sums and Products of Matrices 260—252
143. Associated Determinant 252
144. Null Spaces of Matrices 252—254
[45. Latent Points 254—255
[46L Semi-Latent Regions 256
147. The Identical Equation 256—257
48. The Latent Region of a Repeated Latent Root 257—258
L49. The First Species of Semi-Latent Regions 258—259
50. The ^igher Species of Semi-Latent Regions 259—261
51. The Identical Equation 261
52. The Vacuity of a Matrix 261—262
153. Symmetrical Matrices 262—265
[54. Symmetrical Matrices and Supplements 265 — 267
[55. Skew Matrices 267—269
BOOK V.
EXTENSIVE MANIFOLDS OF THREE DIMENSIONS.
CHAPTER I.
Systems of Forces.
156. Non-metrical Theory of Forces 273—274
157. Recapitulation of Formula 274—275
158. Inner Multiplication 275—276
159. Elementary Properties of a Single Force 276
160. Elementary Properties of Systems of Forces 276 — 277
161. Condition for a Single Force 277
162. Ck>njugate lines 277—278
163. Null Lines, Planes and Points 278
164. Properties of Null Lines 279—280
165. Lines in Involution 280—281
166. Reciprocal Systems 281—282
167. Formuka for Systems of Forces 282—283
XX CONTENTS.
CHAPTER 11.
Groups of Systems of Forces.
AKT. PAQKS
168. Specifications of a Group 284 — 285
169. Systems Reciprocal to Qroups 285
170. Common Null Lines and Director Forces 286
171. Quintuple Groups 286—287
172. Quadruple and Dual Groups . 287—290
173. Anharmonic Ratio of Systems 290 — 292
174. Self-Supplementary Dual Groups 292—294
176. Triple Groups 295—298
176. Conjugate Sets of Systems in a Triple Group 298 — 299
CHAPTER III.
Invariants of Groups.
177. Definition of an Invariant 300
178. The Null Invariants of a Dual Group 300
179. The Harmonic Invariants of a Dual Group 301—302
180. Further Properties of Harmonic Invariants 302 — 303
181. Formulad connected with Reciprocal Systems 303—304
182. Systems Reciprocal to a Dual Group 304
183. The Pole and Polar Invariants of a Triple Group .... 305—306
184. Conjugate Sets of Systems and the Pole and Polar Invariants 306—307
185. Interpretation of P (^) and P (Z) 307—308
186. Relations between Conjugate Sets of Systems 308—310
187. The Conjugate Invariant of a Triple Group 310—312
188. Transformations of Q (p, />) and (? (P, P) 312—315
CHAPTER IV.
Matrices and Forces.
189. Linear Transformations in Three Dimensions 316 — 317
190. Enumeration of Types of Latent and Semi-Latent Regions . 317 — 321
191. Matrices and Forces 322—323
192. Latent Systems and Semi-Latent Groups 323—326
193. Enumeration of Types of Latent Systems and Semi-Latent Grou^xs . 326 — 338
194 Transformation of a Quadric into itself 338—339
195. Direct Transformation of Quadrics 339—342
196. Skew Transformation of Quadrics 342—346
Note 346
CONTENTS. Xxi
BOOK VL
THEORY OF METRICS.
CHAPTER I.
Thbx)ry of Distance.
A^RT. PAGES
197. Axioms of Distance 349—350
198. Congruent Ranges of Points 350—351
199. Cayle/s Theory of Distance 361 353
200. Klein's Theorem 353 — 354
201. Comparison with the Axioms of Distance 354
202. Spatial Manifolds of Many Dimensions 354 — 355
203. Division of Space . 355 356
204. Elliptic Space 356
205. Polar Form 356 — 368
206. Length of Intercepts in Polar Form 368 — 361
207. Antipodal Form 361—362
208. Hyperbolic Space 362—363
209. The Space Constant 363—364
210. Law of Intensity in Elliptic and Hyperbolic Geometry 364—366
211. Distances of Planes and of Subregions 365—367
212. Parabolic Geometry 367 — 368
213. Law of Intensity in Parabolic Geometry 368 — 369
Historical Note 369—370
CHAPTER IL
Elliptic Geometry.
214. Introductory 371
215. Triangles 371—373
216. Further Formulue for Triangles 374—375
217. Points inside a Triangle 375 — 376
218. Oval Quadrics 376—378
219. Pmrther Properties of Triangles 378 — 379
220. Planes One-sided 379—382
221. Angles between Planes 382
222. Stereometrical Triangles 382 — 383
223. Perpendiculars 383—386
224. Shortest Distances from Points to Planes 385—386
226. Common Perpendicular of Planes 386
226. Distances from Points to Subregions 387 — 388
227. Shortest Distances between Subregions 388 — 391
228. Spheres 391—396
229. Pajrallel Subregions 397—398
XXU CONTENTS.
CHAPTER III.
Extensive Manifolds and Elliptic Geometry.
ABT. PAOBB
230. Intensities of Forces 399_400
231. Relations between Two Forces 400—401
232. Axes of a System of Forces 401 — 404
233. Non-Axal Systems of Forces 404
234. Parallel Lines 404—406
235. Vector Systems 406 — 407
236. Vector Systems and Parallel Lines 407—408
237. Further Properties of Parallel Lines 409-^11
238. Planes and Parallel Lines 411-^13
CHAPTER IV.
Hyperbolic Geometry.
239. Space and Anti-Space 414
240. Intensities of Points and Planes 415—416
241. Distances of Points 416—417
242. Distances of Planes 417—418
243. Spatial and Anti-spatial Lines 418—419
244. Distances of Subregions 419
245. Geometrical Signification 420
246. Poles and Polars 420—422
247. Points on the Absolute 422
248. Triangles 422—424
^ 249. Properties of Angles of a Spatial Triangle 424 — 425
I 250. Stereometrical Triangles 425—426
[ 251. Perpendiculars 426 — 427
I 252. The Feet of Perpendiculars 427—428
253. Distance between Planes 428 — 429
I 254. Shortest Distances 429—430
I 255. Shortest Distances between Subregions 430 — 433
256. Rectangular Rectilinear Figures 433 — 436
^ 257. Parallel Lines 436—438
258. Parallel Planes 439—440
CHAPTER V.
Hyperbolic Qeometry (continiied).
269. The Sphere 441—444
260. Intersection of Spheres 444 — 447
261. Limit-Surfaces 447—448
262. Qreat Circles on Spheres 448—451
\
CONTENTS. XXlll
ART. PAQES
263. Surfaces of Equal Distance from Subregions 451
264. Intensities of Forces 452
265. Relations between Two Spatial Forces 452 — 454
266. Central Axis of a System of Forces 454--455
267. Non-Axal Systems of Forces 455
CHAPTER VI.
Kinematics in Three Dimensions.
268. Congruent Transformations 456 — 458
269. Elementary Formul® 453^459
270. Simple Geometrical Properties 459 — 460
271. Translations and Rotations 460 — 462
272. Locus of Points of Equal Displacement 462 — 463
273. Equivalent Sets of Congruent Transformations 463
274. Commutative Law 464
275. Small Displacements 464 — 465
276. Small Translations and Rotations 465 — 466
277. Associated System of Forces 466
278. Properties deduced from the Associated System 467 — 468
279. Work 468—469
280. Characteristic Lines 470
281. Elliptic Space 470—471
282. Surfaces of Equal Displacement 472
283. Vector Transformations 472
284. Associated Vector Systems of Forces 473
285. Successive Vector Transformations 473 — 476
286. Small Displacements 476—477
CHAPTER VII.
CuRVKs AND Surfaces.
287. Curve Lines 478—479
288. Ciurvature and Torsion 479—481
289. Planar Formulas 481—482
290. Velocity and Acceleration 482 — 484
291. The Circle 484—487
292. Motion of a Rigid Body 487-488
293. Gauss' Curvilinear Coordinates 488-489
294. Curvature of Surfaces 489—490
295. Lines of Curvature 490—493
296. Meunier^s Theorem 493
297. Normals 493—494
298. Curvilinear Coordinates 494
299. Limit-Surfaces 494—495
XXIV CONTENTS.
CHAPTER VIIL
Transition to Parabolic Geometry.
ABT. PAGES
300. Parabolic Geometry 4d6
301. Plane Equation of the Absolute 496 — 498
302. Intensities 498—499
303. Congruent Transformations 500 — 502
%
t
BOOK VII.
APPLICATION OF THE CALCULUS OF EXTENSION TO
GEOMETRY.
CHAPTER I.
Vectors.
304. Introductory 505—606
305. Points at Infinity 506—507
306. Vectors 507—508
307. Linear Elements 508—509
308. Vector Areas 509—511
309. Vector Areas as Carriers 511
310. Planar Elements 512—513
311. Vector Volumes 513
312. Vector Volumes as Carriers 513 — 514
313. Product of Four Points 514
314. Point and Vector Factors 514 — 515
315. Interpretation of Formul® 515 — 516
316. Vector Formul® 516
317. Operation of Taking the Vector 516—518
818. Theory of Foitses 518—520
319. Graphic Statics 520—522
Note 522
CONTENTS.
XXV
CHAPTER II.
Vectors (continued).
f
AET.
320. Supplements ....
321. Rectangular Normal Systems .
322. Imaginaiy Self-Normal Sphere
323. Real Self-Normal Sphere
324. Qeometrical Formulas
325. Taking the Flux
326. Flux Multiplication .
327. Geometrical Formulae
328. The Central Axis .
329. Planes containing the Central Axis
330. Dual Groups of Systems of Forces
331. Invariants of a Dual Group .
332. Secondary Axes of a Dual Group .
333. The Cylindroid ....
334. The Harmonic Invariants
335. Triple Groups
336. The Pole and Polar Invariants
337. Equation of the Associated Quadric
338. Normals
339. Small Displacements of a Rigid Body
340. Work
PAOES
523—524
524
524—525
526—526
526—527
627—528
628
529
529—530
530
530—531
531
531—532
532—533
533
533—534
534—535
535
535 536
636—537
637-538
CHAPTER IIL
Curves and Surfaces.
341. Curves
342. Osculating Plane and Normals
343. Acceleration ....
344. Simplified FormulsB .
345. Spherical Curvature
346. Locus of Centre of Ciurvature
347. Gauss' Curvilinear Co-ordinates
348. Curvature
349. Lines of Curvature
350. Dupin's Theorem
361. Ruler's Theorem
352 Meunier's Theorem
Note
539
540
540
541
641—542
542—643
643—644
644—645
545—646
646—547
647
547
547
r
XXVI CONTENTS.
CHAPTER IV.
\ Pure Vector Formula.
I
ABT. PAQB8
I
363. Introductory 548—549
354. Lengths and Areas 549
355. FormulfiB 549—550
356. The Origin 550
357. New Convention 550 — 551
358. System of Forces 551
359. Kinematics 551—652
360. A Continuously Distributed Substance 552 — 554
361. Hamilton's Differential Operator 564—555
362. Conventions and Formulae 555 — 557
363. Polar Co-ordinates 557—568
364. Cylindrical Co-ordinates 558—560
365. Orthogonal Curvilinear Co-ordinates 560 — 562
366. Volume, Surface, and Line Integrals 562
367. The Equations of Hydrodynamics 562—563
368. Moving Origin 563—565
369. Transformations of Hydrodynamical Equations 565
370. Vector Potential of Velocity 566—566
371. Curl Filaments of Constant Strength 567—569
372. Carried Functions 569—570
373. Clebsch's Transformations 670—572
374. Flow of a Vector 572—573
Note 573
Note on Orastmann 573 — 575
Index 576—586
i
BOOK I.
PRINCIPLES OF ALGEBRAIC SYMBOLISM.
w. 1
i
V
^
r
CHAPTER I.
On the nature of a Calculus.
1. Signs. Words, spoken or written, and the symbols of Mathematics
ai^ alike signs. Signs have been analysed* into (a) suggestive signs, I
(j8) expressive signs, (7) substitutive signs.
A suggestive sign is the most rudimentary possible, and need not be
dwelt upon here. An obvious example of one is a knot tied in a band-
kerchief to remind the owner of some duty to be performed.
In the use of expressive signs the attention is not fixed on the sign itself
but on what it expresses; that is to say, it is fixed on the meaning conveyed
by the sign. Ordinary language consists of groups of expressive signs, its
primary object being to draw attention to the meaning of the words
employed. Language, no doubt, in its secondary uses has some of the
characteristics of a system of substitutive signs. It remedies the inability
of the imagination to bring readily before the mind the whole extent of
complex ideas by associating these ideas with familiar sounds or marks ;
and it is not always necessary for the attention to dwell on the complete
meaning while using these Sjnoibols. But with all this allowance it remains
true that language when challenged by criticism refers us to the meaning
and not to the natural or conventional properties of its symbols for an
explanation of its processes.
A substitutive, sign is such that in thought it takes the place of that for
which it is substituted. A counter in a game may be such a sign : at the
end of the game the counters lost or won may be interpreted in the form of
money, but till then it may be convenient for attention to be concentrated
on the counters and not on their signification. The signs of a Mathematical
Calculus are substitutive signs.
The difference between words and substitutive signs has been stated
thus, 'a word is an instrument for thinking about the meaning which it
* Cf. Stout, 'Thought and Language,* 3/tit<7, April, 1891, repeated in the same author's
Analytic Ptyehology^ (1896), oh. x. § 1: cf. also a more obscure analysis to the same e£Feot by
C. S. Peirce, Proe, of the American Academy of, Arts and Scittncet^ 1867, Vol. vii. p. 294.
1—2
I
4 ON THE NATURE OF A CALCULUS. [CHAP. L
expresses ; a substitute sign is a means of not thinking about the meaning
which it symbolizes*.' The use of substitutive signs in reasoning is to
economize thought.
2. Definition of a Calculus. In order that reasoning may be con-
ducted by means of substitutive signs, it is necessary that rules be given for
the manipulation of the signs. The rules should be such that the final state
of the signs after a series of operations according to rule denotes, when the
signs are interpreted in terms of the things for which they are substituted,
a proposition true for the things represented by the signs.
The art of the manipulation of substitutive signs according to 6xed rules,
and of the deduction therefix^m of true propositions is a Calculua
We may therefore define a sign used in a Calculus as 'an arbitrary
mark, having a fixed interpretation, and susceptible of combination with
other signs in subjection to fixed laws dependent upon their mutual
interpretation f.*
The interpretation of any sign used in a series of operations must be
fixed in the sense of being the same throughout, but in a certain sense it may
be ambiguous. For instance in ordinary Algebra a letter x may be used
in a series of operations, and x may be defined to be any algebraical
quantity, without further specification of the special quantity chosen.
Such a sign denotes any one of an assigned class with certain un-
ambiguously defined characteristics. In the same series of operations the
sign must always denote the same member of the class ; but as far as any
explicit definitions are concerned any member will do.
When once the rules for the manipulation of the signs of a calculus
are known, the art of their practical manipulation can be studied apart
from any attention to the meaning to be assigned to the signs. It is
obvious that we can take any marks we like and manipulate them
according to any rules we choose to assign. It is also equally obvious that
in general such occupations must be Mvolous. They possess a serious
scientific value when there is a similarity of type of the signs and of the
rules of manipulation to those of some calculus in which the marks used
axe substitutive signs for things and relations of thinga The comparative
study of the various forms produced by variation of rules throws light
on the principles of the calculus. Furthermore the knowledge thus gained
gives fisMjility in the invention of some significant calculus designed to
facilitate^ reasoning with respect to some given subject.
It enters therefore into the definition of a calculus properly so called
that the marks used in it are substitutive signs. But when a set of marks
and the rules for their arrangements and rearrangements are analogous to
* Of. stout, * Thought and Language,' MUid, April, 1S91.
f Boole, Laws of Thought, Ch. ii.
2, 3] DEFINITION OF A CALCULUS. 5
those of a significant calculus so that the study of the allowable forms of
their arrangements throws light on that of the calculus, — or when the
marks and their rules of arrangement are such as appear likely to receive
an interpretation as substitutive signs or to facilitate the invention of a
true calculus, then the art of arranging such marks may be called — by
an extension of the term — ^an uninterpreted calculus. The study of such
a calculus is of scientific value. The marks used in it will be called signs
or symbols as are those of a true calculus, thus tacitly suggesting that
there is some unknown interpretation which could be given to the
calculus.
3. Equivalence. It is necessary to note the form in which propositions
occur in a calculu& Such a form may well be highly artificial from some
points of view, and may yet state the propositions in a convenient form for
the eliciting of deductions. Furthermore it is not necessary to assert that
the form is a general form into which all judgments can be put by the aid
of some torture. It is sufficient to observe that it is a form of wide appli-
cation.
In a calculus of the type here considered propositions take the form
of assertions of equivalence. One thing or &ct, which may be complex and
involve an inter-related group of things or a succession of facts, is asserted
to be equivalent in some sense or other to another thing or fact.
Accordingly the sign = is taken to denote that the signs or groups of
signs on either side of it are equivalent, and therefore symbolize things
which are so far equivalent. When two groups of symbols are connected by
this sign, it is to be understood that one group may be substituted for the
other group whenever either occurs in the calculus under conditions for
which the assertion of equivalence holds good.
The idea of equivalence requires some explanation. Two things are
equivalent when for some purpose they can be used indifferently. Thus the
equivalence of distinct things implies a certain defined purpose in view, a
certain limitation of thought or of action. Then within this limited field
no distinction of property exists between the two things.
As an instance of the limitation of the field of equivalence consider
an ordinary algebraical equation, /(a?, y) = 0. Then in finding ^ by the
formula, ;r^ = — ^ / ^ » ^^ ^^y ^^^ substitute 0 for / on the right-hand
side of the last equation, though the equivalence of the two symbols has been
asserted in the first equation, the reason being that the limitations under
which /= 0 has been asserted are violated when / undergoes partial dif-
ferentiation.
The idea of equivalence must be carefully distinguished from that of
\
6
ON THB NATURE OF A CALCULUS.
[chap. I.
i
mere identity*. No investigations which proceed by the aid of propositions
merely asserting identities such as il is ^, can ever result in anything but
barren identities^. Equivalence on the other hand implies non-identity
as its general case. Identity may be conceived as a special limiting
case of equivalence. For instance in arithmetic we write, . 2 -I- 8 » 3 -f 2.
This means that, in so far as the total number of objects mentioned, 2 -f 3
and 3 + 2 come to the same number, namely 5. But 2 -f 3 and 3 + 2 are
not identical ; the order of the symbols is different in the two combinations,
and this difference of order directs different processes of thought. The
importance of the equation arises from its assertion that these different
processes of thought are identical as far as the total number of things
thought of is concerned.
From this arithmetical point of view it is tempting to define equivalent
things as being merely different ways of thinking of the same thing as it
exists in the external world. Thus there is a certain aggregate, say of 5
things, which is thought of in different ways, as 2 + 3 and as 3 + 2. A
sufficient objection to this definition is that the man who shall succeed in
stating intelligibly the distinction between himself and the rest of the world
will have solved the central problem of philosophy. As there is no
universally accepted solution of this problem, it is obviously undesirable to
assume this distinction as the basis of mathematical reasoning.
Thus from another point of view all things which for any purpose can be
conceived as equivalent form the extension (in the logical sense) of some uni-
versal conception. And conversely the collection of objects which together form
the extension of some universal conception can for some purpose be treated
as equivalent. So 6 = 6^ can be interpreted as symbolizing the fact that the
two individual things b and b' are two individual cases of the same general
conception B\. For instance if b stand for 2 + 3 and b' for 3 + 2, both b and
b' are individual instances of the general conception of a group of five things.
The sign = as used in a calculus must be discriminated from the logical
copula ' is.' Two things b and b' are connected in a calculus by the sign =,
so that b = b\ when both b and V possess the attribute B. But we may not
translate this into the standard logical form, b is b\ On the contrary, we
say, b ia By and b' is B; and we may not translate these standard forms
of formal logic into the symbolic form, 6 = B, 6' = B ; at least we may not do
so, if the sign = is to have the meaning which is assigned to it in a calculua
It is to be observed that the proposition asserted by the equation, b=b\
consists of two elements ; which for the sake of distinctness we will name,
and will call respectively the * truism 'and the ' paradox.' The truism is the
partial identity of both b and b\ their common J3-nes& The paradox is the
* Cf. Lotze, LogiCf Bk. i. Gh. n. Art. 64.
t Gf. Bradley, PrifieipU$ of Logic, Bk. i. Gh. ▼.
X Ibid, Bk. n. Pt. i. Gh. iv. Art. 3 (p).
4] EQUIVALENCE. 7
distinction between b and b\ so that b is one thing and 6' is another thing : |
and these things, as being different, must have in some relation diverse
properties. In assertions of equivalence as contained in a calculus the- truism
is passed over with the slightest possible attention, the main stress being laid /
on the paradox. Thus in the equation 2 + 3 = 3 + 2, the fact that both sides '
represent a common five-ness of number is not even mentioned explicitly.
The sole direct statement is that the two different things 3 + 2 and 2 + 3 ^
are in point of number equivalent.
The reason for this unequal distribution of attention is easy to under-
stand. In order to discover new propositions asserting equivalence it is ^
requisite to discover easy marks or tests of equivalent things. These
tests are discovered by a careful discussion of the truism, of the common
^-ness of b and b'. But when once such tests have been elaborated, we may
drop all thought of the essential nature of the attribute B, and simply
apply the superficial test to b and 6' in order to verify 6 = 6'. Thus in
order to verify that thirty-seven times fifty-six is equal to fifty-six times
thirty-seven, we may use the entirely superficial test applicable to this case
that the same &ctors ai*e mentioned as multiplied, though in different
order.
This discussion leads us at once to comprehend the essence of a calculus
of substitutive signs. The signs are by convention to be considered equiva- r
lent when certain conditions hold. And these conditions when inter-
preted imply the fulfilment of the tests of equivalence.
Thus in the discussion of the laws of a calculus stress is laid on the
truism, in the development of the consequences on the paradox.
4 Operations. Judgments of equivalence can be founded on direct
perception, as when it is judged by direct perception that two different pieces
of stuff match in colour. But the judgment may be founded on a knowledge
of the respective derivations of the things judged to be equivalent fix)m other
things respectively either identical or equivalent. It is this process of
derivation which is the special province of a calculus. The derivation of
a thing p from things a, 6, c, ... , can also be conceived as an operation on
the things a, 6, c, ... , which produces the thing p. The idea of derivation j^
includes that of a series of phenomenal occurrences. Thus two pieces of stuff J
may be judged to match in colour because they were dyed in the same
dipping, or were cut from the same piece of stuff. But the idea is more
general than that of phenomenal sequence of events: it includes purely
logical activities of the mind, as when it is judged that an aggregate of five
things has been presented to the mind by two aggregates of three things and
of two things respectively. Another example of derivation is that of two
propositions a and 6 which are both derived by strict deductive reasoning
from the sfime propositions c, d, and e. The two propositions are either both
8 ON THE NATURE OF A CALCULUS. [CHAP. I.
proved or both unproved according as c, d, and e are granted or disputed.
Thus a and h are so fai' equivalent. In other words a and 6 may be considered
as the equivalent results of two operations on c, d and e.
The words operation, combination, derivation, and synthesis will be used
to express the same general idea, of which each word suggests a somewhat
specialized form. This general idea may be defined thus : A thing a will be
said to result from an operation on other things, c, d, e, etc., when a is
presented to the mind as the result of the presentations of c, d and e, etc.
under certain conditions; and these conditions are phenomenal events or
mcDtal activities which it is convenient to separate in idea into a group by
themselves and to consider as defining the nature of the operation which is
performed on c, d, e, etc.
Furthermore the fact that c, d, e, etc. are capable of undergoing a certain
operation involving them all will be considered as constituting a relation
between c, rf, «, etc.
Also the fact that c is capable of undergoing an operation of a certain
general kind will be considered as a property of c. Any additional speciali-
zation of the kind of operation or of the nature of the result will be considered
as a mode of that property.
6. Substitutive Schemes. Let a, a', etc., 6, h\ etc., z, /, etc.,
denote any set of objects considered in relation to some common property <
which is symbolized by the use of the italic alphabet of letters. The
common property may not be possessed in the same mode by different
members of the set. Their equivalence, or identity in relation to this property,
is symbolized by a literal identity. Thus the fiek^t that the things a and m
are both symbolized by letters from the italic alphabet is here a sign that
the things have some property in common, and the fact that the letters
a and m' are different letters is a sign that the two things possess this
common property in different modes. On the other hand the two things
a and a' possess the common property in the same mode, and as far as
this property is concerned they are equivalent. Let the sign = express
equivalence in relation to this property, then a = a\ and m = m\
Let a set of things such as that described above, considered iu relation
to their possession of a common property in equivalent or in non-equivalent
modes be called a scheme of things ; and let the common property of which
the possession by any object marks that object as belonging to the scheme ^
be called the Determining Property of the Scheme. Thus objects belonging ^
to the same scheme are equivalent if they possess the determining property
in the same mode.
Now relations must exist between non-equivalent things of the scheme
which depend on the differences between the modes in which they possess
the determining property of the scheme. In consequence of these relations
<0aummmimmmit^
5, 6] SUBSTITUTIVE SCHEMES. 9
from things a, 6, c, etc. of the scheme another thing m of the scheme can be
derived by certain operations. The equivalence, m = m', will exist between
m and w!^ if m and w! are derived from other things of the scheme by
operations which only differ in certain assigned modes. The modes in which
processes of derivation of equivalent things m and w! from other things of
the scheme can differ without destro}dng the equivalence of m and m' will be
called the Characteristics of the scheme.
Now it may happen that two schemes of things — with of course different
determining properties — have the same characteristica Also it may be
possible to establish an unambiguous correspondence between the things
of the two schemes, so that if a, a\ 6, etc., belong to one scheme and
a, a, fi, etc., belong to the other, then a corresponds to a, a' to a\ b to fi
and so on. The essential rule of the correspondence is that if in one scheme
two things, say a and a\ are equivalent, then in the other scheme their
corresponding things a and a! are equivalent. Accordingly to any process
of derivation in the italic alphabet by which m is derived from a, 6, eta
there must correspond a process of derivation in the Qreek alphabet by
which /A is derived from a, fi, etc.
In such a case instead of reasoning with respect to the properties of one
scheme in order to deduce equivalences, we may substitute the other
scheme, or conversely; and then transpose at the end of the argument.
This device of reasoning, which is almost universal in mathematics, we will
call the method of substitutive schemes, or more briefly, the method of
substitution.
These substituted things belonging to another scheme are nothing else
than substitutive signs. For in the use of substituted schemes we cease to
think of the original scheme. The rule of reasoning is to confine thought
to those properties, previously determined, which are shared in common with
the original scheme, and to interpret the results from one set of things into
the other at the end of the argument.
An instance of this process of reasoning by substitution is to be found
in the theory of quantity. Quantities are measured by their ratio to an
arbitrarily assumed quantity of the same kind, called the unit. Any set of
quantities of one kind can be represented by a corresponding set of quantities
of any other kind merely in so far as their numerical ratios to their unit are
concerned. For the representative set have only to bear the same ratios
to their unit as do the original set to their unit.
6. Conventional Schemes. The use of a calculus of substitutive
signs in reasoning can now be explained.
Besides using substitutive schemes with naturally suitable properties,
we may by convention assign to arbitrary marks laws of equivalence which
are identical with the laws of equivalence of the originals about which we
I
-*
10 ON THE NATURE OP A CALCULUS. [CHAP. I.
desire to reason. The set of marks may then be considered as a scheme
of things with properties assigned by convention. The determining property
of the scheme is that the marks are of certain assigned sorts arranged
in certain types of sequence. The characteristics of the scheme are
the conventional laws by which certain arrangements of the marks in
sequence on paper are to be taken as equivalent. As long as the marks
are treated as mutually determined by their conventional properties,
reasoning concerning the marks will hold good concerning the originals
for which the marks are substitutive signs. For instance in the employ-
ment of the marks a?, y, -f , the equation, a? + y = y + ar, asserts that a
certain union on paper of x and y possesses the conventional quality that
the order of x and y is indifferent. Therefore any union of two things
with a result independent of any precedence of one thing before the other
possesses so far properties identical with those of the union above
set down between x and y. Not only can the reasoning be transferred
from the originals to the substitutive signs, but the imaginative thought
itself can in a large measure be avoided. For whereas combinations of the
original things are possible only in thought and by an act of the imagi-
nation, the combinations of the conventional substitutive signs of a calculus
are physically made on paper. The mind has simply to attend to the rules
for transformation and to use its experience and imagination to suggest
likely methods of procedure. The rest is merely phjrsical actual inter-
change of the signs instead of thought about the originals.
A calculus avoids the necessity of inference and replaces it by an ex-
ternal demonstration, where inference and external demonstration are
to be taken in the senses assigned to them by F. H. Bradley*. In this
connexion a demonstration is to be defined as a process of combining a
complex of facts, the data, into a whole so that some new fact is evident.
Inference is an ideal combination or construction within the mind of the
' reasoner which results in the intuitive evidence of a new fact or relation
between the data. But in the use of a calculus this process of combina-
tion is externally performed by the combination of the concrete symbols,
with the result of a new fact respecting the symbols which arises for sensuous
perception f. When this new fact is treated as k symbol carrying a 'X
meaning, it is found to mean the fact which would have been intuitively
evident in the process of inference.
7. Uninterpretable Forms. The logical diflScultyJ involved in the
use of a calculus only partially interpretable can now be explained. The
♦ Cf. Bradley, Principles of Logic, Bk ii. Pt i. Oh. iii.
I t Cf. C. S. Peiroe, Amer, Joum. of Math, VoL vn. p. 1S2 : ' Ab for algebra, the very idea of
; the art is that it presents formolaB which can be manipalated, and that by observing the effects
I of BQoh manipulation we find properties not otherwise to be discovered/
X Cf. Boole, Lam of Thought, Ch. v. § 4.
4
<
7] UNINTERPRETABLE FORMS. 11
discussion of this great problem in its application to the special case of
(— 1)* engaged the attention of the leading mathematicians of the first half
of this century, and led to the development on the one hand of the Theory
of Functions of a Complex Variable, and on the other hand of the science
here called Universal Algebra.
The difficulty is this : the symbol (—1)' is absolutely without meaning
when it is endeavoured to interpret it as a number; but algebraic trans-
formations which involve the use of complex quantities of the form a + 6t,
where a and 6 are numbers and i stands for the above symbol, yield pro-
positions which do relate purely to number. As a matter of faet the pro-
positions thus discovered were found to be true propositions. The method
therefore was trusted, before any explanation was forthcoming why algebraic
reasoning which had no intelligible interpretation in arithmetic should
give true arithmetical results.
The difficulty was solved by observing that Algebra does not depend on \
Arithmetic for the validity of its laws of transformation. If there were »
such a dependence, it is obvious that as soon as algebraic expressions
are arithmetically unintelligible all laws respecting them must lose their
validity. But the laws of Algebra, though suggested by Arithmetic, do
not depend on it. They depend entirely on the convention by which it is
stated that certain modes of grouping the symbols are to be considered as
identical. This assigns certain properties to the marks which form the symbols
of Algebra. The laws regulating the manipulation of the algebraic symbols
are identical with those of Arithmetic. It follows that no algebraic theorem
can ever contradict any result which could be arrived at by Arithmetic ; for
the reasoning in both cases merely applies the same general laws to diffei-ent
classes of things. If an algebraic theorem is interpretable in Arithmetic,
the corresponding arithmetical theorem is therefore true. In short when
once Algebra is conceived as an independent science dealing with the re-
lations of certain marks conditioned by the observance of certain conventional
laws, the difficulty vanishes. If the laws be identical, the theorems of the
one science can only give results conditioned by the laws which also hold
good for the other science ; and therefore these results, when interpretable,
are true.
It will be observed that the explanation of the legitimacy of the use of a
partially interpretable calculus does not depend upon the fact that in another
field of thought the calculus is entirely interpretable. The discovery of an
interpretation undoubtedly gave the clue by means of which the true solution
was arrived at. For the fact that the processes of the calculus were in-
terpretable in a science so independent of Arithmetic as is Geometry at once
showed that the laws of the calculus might have been defined in reference
to geometrical processes. But it was a paradox to assert that a science like
Algebra, which had been studied for centuries without reference to Geometry,
12 ON THE NATURE OF A CALCULUS. [CHAP. I. 7
was after all dependent upon Geometry for its first principles. The step to
the true explanation was then easily taken.
But the importance of the assistance given to the study of Algebra by the
discovery of a complete interpretation of its processes cannot be over-esti-
mated. It is natural to think of the substitutive set of things as assisting
the study of the properties of the originals. Especially is this the case with
a calculus of which the interest almost entirely depends upon its relation to
the originals. But it must be remembered that conversely the originals give
immense aid to the study of the substitutive things or symbols.
The wbole of Mathematics consists in the organization of a series of aids
\ to the imagination in the process of reasoning ; and for this purpose device is
1 piled upon device. No sooner has a substitutive scheme been devised to assist
tin the investigation of any originals, than the imagination begins to use the
^originals to assist in the investigation of the substitutive scheme. In some
connexions it would be better to abandon the conception of originals studied
by the aid of substitutive schemes, and to conceive of two sets of inter-related
things studied together, each scheme exemplifying the operation of the same
general laws. The discovery therefore of the geometrical representation of
the algebraical complex quantity, though unessential to the logic of Algebra,
has been quite essential to the modem developments of the science.
V
CHAPTER II.
Manifolds.
8. Manifolds. The idea of a manifold was first explicitly stated by
Riemann*; Qrassmannf had still earlier defined and investigated a particular
kind of manifold.
Consider any number of things possessing any common property.
That property may be possessed by different things in different modes : let
each separate mode in which the propeiiiy is possessed be called an element.
The aggregate of all such elements is called the manifold of the property.
Any object which is specified as possessing a property in a given mode
corresponds t>o an element in the manifold of that property. The element
may be spoken of as representing the object or the object as representing
the element according to convenience. All such objects may be conceived
as equivalent in that they represent the same element of the manifold.
Various relations can be stated between one mode of a property and
another mode ; in other words, relations exist between two objects, whatever .
other properties they may possess, which possess this property in any two 1
assigned modes. The relations will define how the objects necessarily differ /
IB that they possess this property differently : they define the distinction
between two sorts of the same property. These relations will be called
relations between the various elements of the manifold of the property ; and
the axioms from which can be logically deduced the whole aggregate of such
relations for all the elements of a given manifold are called the characteristics
of the manifold.
The idea of empty space referred to coordinate axes is an example of a
manifold. Each point of space represents a special mode of the common
property of spatiality. The fundamental properties of space expressed in
terms of these coordinates, i.e. all geometiical axioms, form the character-
istics of this manifold.
* Ueher die Hypotheten, welche der Qeometrie zu Orunde liegen, QesammeUe Mathematisehe
Werke ; a translation of this paper is to be found in Clifford's Collected Mathematical Papers,
t Afudefmvngslehre von 1S44.
14 MANIFOLDS. [CUAP. II.
It is the logical deductions from the characteristics of a manifold which
are investigated by means of a calculus. The manifolds of separate proper-
ties may have the same characteristics. In such a case all theorems which
are proved for one manifold can be directly translated so as to apply to
the other. This is only another mode of stating the ideas explained in
Chapter I. §§ 3, 4, 5.
The relation of a manifold of elements to a scheme of things (cf. § 5), is
that of the abstract to the concrete. Consider as explained in § 5 the
scheme of things represented by a, a' etc., 6, V etc., z, / etc. Then
these concrete things are not elements of a manifold. But to such a scheme
a manifold always corresponds, and conversely to a manifold a scheme of
things corresponds. The abstract property of a common ^-ness which makes
the equivalence of a, a\ etc., in the scheme is an element of the manifold
which corresponds to this scheme. Thus the relation of a thing in a scheme
to the corresponding element of the corresponding manifold is that of a
subject of which the element can be predicated. If il be the element
corresponding to a, a! etc., then a \& A^ and a' is A. Thus if we write
2-1-3 = 5 at length, the assertion is seen to be
(l + l) + (l-f-l + l) = l-f 1 + 1 + 1 + 1; •
this asserts that two methods of grouping the marks of the type 1 are
equivalent as far as the common five-ness of the sum on each side.
The manifold corresponding to a scheme is the manifold of the deter-
mining property of the scheme. The cliaiucteristics of the manifold corre-
spond to the characteristics of the scheme.
9. Secondary Properties of Elements. In order to state the
characteristics of a manifold it may be necessaiy to ascribe to objects coiTe-
sponding to the elements the capability of possessing other properties in
addition to that definite property in special modes which the elements
represent. Thus for the purpose of expressing the relation of an element A
of a manifold to the elements B and C it may be necessary to conceive
an object corresponding to A which is either Oi or a,, or a,, where the
suffix denotes the possession of some other property, in addition to the
il-ness of A, in some special mode which is here symbolized by the suffix
chosen. Such a property of an object corresponding to A^ which is necessary
to define the relation of il to other elements of the manifold, is called a
Secondary Property of the element A,
Brevity is gained by considering each element of the manifold, such as A,
as containing within itself a whole manifold of its secondary properties.
Thus with the above notation A stands for any one of A^, -4„ A^ etc., where
the suffix denotes the special mode of the secondary property. Hence the
object O], mentioned above, corresponds to A^, and a, to A^, and so on.
9, 10] SECONDARY PROPERTIES OF ELEMENTS. 15
And the statement of the relation between two elements of the original
manifold, such as A and B, requires the mention of a special A, say A^
and of a special B, say B4.
For example consider the manifold of musical notes conceived as repre-
senting eveiy note so far as it differs in pitch and quality from every other
note. Thus each element is a note of given pitch and given quality. The
attribute of loudness is not an attribute which this manifold represents;
but it is a secondary property of the elements. For consider a tone A and
two of its overtones B and C, and consider the relations o{ A, B, C to & note
P which is of the same pitch as A and which only involves the overtones
B and C. Then P can be described as the pitch and quality of the sound
produced by the simultaneous existence of concrete instances o{ A^ B and C
with certain relative loudnesses. Hence the relation of P to A, B, C requires
the mention of the loudness of each element in order to express it Thus
if -dj, -B,, Ci denote A, B, C with the required ratio of their loudnesses, P
might be expressed as the combination of A^, B^, C4.
The sole secondary property with which in this work we shall be concerned
is that of intensity. Thus in some manifolds each element is to be conceived
as the seat of a possible intensity of any arbitrarily assumed value, and this
intensity is a secondary property necessary to express the various relations of
the elements.
10. Definitions. To partition a manifold is to make a selection
of elements possessing a common characteristic : thus if the manifold
be a plane, a selection may be made of points at an equal distance from a
given point. The selected points then form a circle. The selected elements
of a partitioned manifold form another manifold, which may be called a
submanifold in reference to the original manifold.
Again the common attribute C, which is shared by the selected elements
of the original manifold A, may also be shared by elements of another
manifold B. For instance in the above illustration other points in other
planes may be at the same distance from the given point. We thus arrive at
the conception of the manifold of the attribute C which has common elements
with the manifolds A and B, This conception undoubtedly implies that the
three manifolds A, B and C have an organic connection, and are in fact parts
of a manifold which embraces them all three.
A manifold will be called the complete manifold in reference to its
possible submanifolds ; and the complete manifold will be said to contain its
submanifolds. The submanifolds will be said to be incident in the complete
manifold.
One submanifold may be incident' in more than one manifold. It will
then be called a common submanifold of the two manifolds. Manifolds
will be said to intersect in their common. submanifolds.
16 MANIFOLDS. [CHAP. II.
11. Special Manifolds. A few definitions of special manifolds will
both elucidate the general explanation of a manifold given above and will
serve to introduce the special manifolds of which the properties are dis-
cussed in this work.
A manifold may be called self-constituted when only the properties
which the elements represent are used to define the relations between
elements; that is, when there are no secondary properties.
A manifold may be called extrinsically constituted when secondary
properties have to be used to define these relations.
The manifold of integral numbers is self-constituted, since all relations
of such numbers can be defined in terms of them.
A uniform manifold is a manifold in which each element bears the same
relation as any other element to the manifold considered as a whole.
If such a manifold be a submanifold of a complete manifold, it is not
necessary that each element of the uniform submanifold bear the same
relation to the complete manifold as any other element of that submanifold.
Space, the points being elements, forms a uniform manifold. Again
the perimeter of a circle, the points being elements, forms a uniform mani-
fold. The area of a circle does not form a uniform manifold.
A simple serial manifold is a manifold such that the elements can be
arranged in one series. The meaning of this property is that some determinate
process of deriving the elements in order one &om the other exists (as in the
case of the successive integral numbers), and that starting from some initial
element all the other elements of the manifold are derived in a fixed order by
the successive application of this process. Since the process is determinate
for a simple serial manifold, there is no ambiguity as to the order of suc-
cession of elements. The elements of such a manifold are not necessarily
numerable. A test of a simple serial manifold is that, given any three
elements of the manifold it may be possible to conceive their mutual relations
in such a fashion that one of them can be said to lie between the other
two. If a simple serial manifold be uniform it follows that any element can
be chosen as the initial element.
A manifold may be called a complex serial manifold when all its elements
belong to one or more submanifolds which are simple serial manifolds, but
when it is not itself a simple serial manifold. A surface is such a manifold,
while a line is a simple serial manifold.
Two manifolds have a one to one correspondence* between their elements
if to every element of either manifold one and only one element of the
other manifold corresponds, so that the corresponding elements bear a certain
defined relation to each other.
* The Bubjeot of the correspondence between the elements of manifolds has been inyestigated
by G. Cantor, in a series of memoirs entitled, ' Ueber nnendliche, lineare Panktmannichfaltigkeiten,*
Math. Annalen, Bd. 15, 17, 20, 21, 28, and BorehardVt Journal^ Bd. 77, 84.
J
11] SPECIAL MANIFOLDS. 17
A quantitively defined manifold is such that each element is specified by
a definite number of measurable entities of which the measures for any
element are the algebraic quantities ^, 17, ^, etc., so that the manifold has a
one to one correspondence with the aggregate of sets of simultaneous values
of these variables.
A quantitively defined manifold is a manifold of an algebraic function
when each element represents in some way the value of an algebraic quantity
w for a set of simultaneous values of f, rj, (f, etc., where w is a function of
f> V> (r> ^^^'> ^^ ^^^ sense that it can be constructed by definite algebraic
operations on f, 17, (f, etc., regarded as irresoluble magnitudes, real or
imaginary*.
A quantitively defined manifold in which the elements are defined
by a single quantity f is a simple serial manifold as far as real values of ^ are
concerned. For the elements can be conceived as successively generated
in the order in which they occur as f varies from — x to + x .
If an element of the manifold corresponds to each value of ^ as it varies
continuously through all its values, then the manifold may be called con-
tinuous If some values of f have no elements of the manifold corresponding
to them, then the manifold may be called discontinuous.
A quantitively defined manifold depending on more than one quantity is
a complex serial manifold. For if the quantities defining it f , 17, (f, eta be
put equal to arbitrary functions of any quantity t, so that f =/i (t), rj =/, (t),
etc., then a submanifold is formed which is a quantitively deBned manifold
depending on the single quantity r. This submanifold is therefore a simple
serial manifold. But by properly choosing the arbitrary functions such a
submanifold may be made to contain any element of the complete manifold.
Hence the complete manifold is a complex serial manifold.
The quantitively defined manifold is continuous if an element corresponds
to every set of values of the variables.
A quantitively defined manifold which requires for its definition the
absolute values (as distinct from the ratios) of v variables is said to be of 1/
dimensions.
A continuous quantitively defined manifold of v dimensions may also be
called a i/-fold extended continuous manifold f.
♦ Cf. Forsyth, Theory of Functiotu, Ch. i. §§ 6, 7.
t Cf. Riemann, loo. eit. section i. § 2.
W.
CHAPTER ni.
Principles of Universal Algebra.
12. Introductory. Universal Algebra is the name applied to that
calculus which symbolizes general operations, defined later, which are called
Addition and Multiplication. There are certain general definitions which
hold for any process of addition and others which hold for any process of
multiplication. These are the general principles of any branch of Universal
Algebra. These principles, which are few in number, will be considered in the
present chapter. But beyond these general definitions there are other special
definitions which define special kinds of addition or of multiplication. The
development and comparison of these special kinds of addition or of multipli-
. cation form special branches of Universal Algebra. Each such branch will be
called a special algebraic calculus, or more shortly, a special algebra, and
the more important branches will be given distinguishing names. Ordinary
algebra will, when there is no risk of confusion, be called simply algebra ;
but when confusion may arise, the term ordinary will be prefixed.
13. Equivalence. It has been explained in § 3 that the idea of
equivalence requires - special definition for any subject-matter to which it
is applied. The definitions of the processes of addition and multiplication do
carry with them this required definition of equivalence as it occurs in the
field of Universal Algebra. One general definition holds both for addition
and multiplication, and thus through the whole field of Universal Algebra.
This definition may be fi-amed thus: In any algebraic calculus only one
recognized type of equivalence exists.
The meaning of this definition is that if two symbols a and a' be equivalent
in that sense which is explicitl}' recognized in some algebraic calculus by the
use of the symbol =, then either a or a' may be used indifferently in any series
of operations of addition or multiplication of the type defined in that
calculus.
This definition is so far from being obvious or necessary for any symbolic
calculus, that it actually excludes from the scope of Universal Algebra the
12 — 14] EQUIVALENCE. 19
Differential Calculus, excepting limited parts of it. For if /(^, y) be a
function of two independent variables x and y, and the equivalence
f(x, y) = 0, be asserted, then ^/{x, y) and ^f(x, y) are not necessarily
7^ 7i
zero, whereas =- 0 and r- 0 are necessarily zero. • Hence the symbols f{x, y)
and 0 which are recognized by the sign of equality as equivalent according
to one type of equivalence are not equivalent when submitted to some
operations which occur in the calculus.
14. Principles of Addition. The properties of the general operation
termed addition will now be gradually defined by successive specifications.
Consider a group of things, cx)ncrete or abstract, material things or merely
ideas of relations between other things. Let the individuals of this group be
denoted by letters a,b ... z. Let any two of the group of things be capable
of a synthesis which results in some third thing.
Let this S3ni thesis be of such a nature that all the properties which are i
attributed to any one of the original group of things can also be attributed
to this result of the synthesis. Accordingly the resultant thing belongs to
the original group.
Let the idea of order between the two things be attributable to their
synthesis. Thus if a and b be the two things of which the synthesis
is being discussed^ orders as between a first or b first can be attributed to
this synthesis. Also let only ttuo possible alternative orders as between a
and b be material, so as to be taken into explicit consideration when judging
that things are or are not equivalent.
Let the result of the synthesis be unambiguous, in the sense that all
possible results of a special synthesis in so &r as the process is varied by
the variation of non-apparent details are to be equivalent. It is to be
noted in this connection that the properties of the synthesis which are
explicitly mentioned cannot be considered as necessarily defining its nature
unambiguously. The present assumption therefore amounts to the state- ^
ment that the same words (or symbols) are always to mean the same thing,
at leant in eveiy way which can affect equivalence.
This process of forming a synthesis between two things, such as a and 6,
and then of considering a and b, thus united, as a third resultant thing, may
be symbolized by a /> 6*. Here the order is sjrmbolized by the order in which
a and b are mentioned ; accordingly a ^b and b^a symbolize two different
things. Then by definition the only question of order as between a and b
which can arise in this synthesis is adequately symbolized. Also a ^b
whenever it occurs must always mean the same thing, or at least stand for
some one of a set of equivalent things.
* Of. Grassniann, Aundehmtngnlehre von 1844, Preface.
2—2
20 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. III.
Further a /^ 6 is by assumption a thing capable of the same synthesis with
any other of the things a, t, ... u\ Accordingly we may write
p ^ (a ^b) and (a ^b) ^p
to represent the two possible syntheses of the type involving p and a ob.
The bracket is to have the usual meaning that the synthesis within the
bracket is to be performed first and the resultant thing then to be combined
as the symbols indicate.
According to the convention adopted here the symbol a '^ 6 is to be
read from left to right in the following manner: a is to be considered as
given first, and b as joined on to it according to the manner prescribed
by the symbol ^ . Thus (a '^b) ^p means that the result of a /> 6 is first
obtained and then p is united to it. But a ^ 6 is obtained by taking a
and joining 6 on to it. Thus the total process may equally well be defined
by a ^b '^p. Hence, since both its right-hand and left-hand sides have
been defined to have the same meaning, we obtain the equation
a ^ b ^ p = {a o b) ^ p.
Definition. Let any one of the symbols, either a single letter or a com-
plex of letters, which denotes one of the group of things capable of this
synthesis be called a term. Let the symbol ^ be called the sign of the
operation of this synthesis.
It will be noticed that this synthesis has essentially been defined as a
synthesis between two terms, and that when three terms such as a, 6, j>, are
indicated as subjects of the synthesis a sequence or time-order of the opera-
tions is also unambiguously defined. Thus in the sjmtheses (a ^ 6) ^ j> there
are two separate ideas of order symbolized ; namely^ the determined but
unspecified idea of order of synthesis as between the two terms which is
involved by hypothesis in the act of synthesis, and further the sequence of the
two successive acts of synthesis, and this time-order involves the sequence
in which the various terms mentioned are involved in the process. Thus
ar\b rsp and jp '^'(a '^ 6) both involve that the synthesis a ^ 6 is to be first
performed and then the synthesis of a ^ 6 and p according to the special
order of synthesis indicated.
In the case of three successive acts of sjrnthesis an ambiguity may arise.
Consider the operations indicated in the symbols
a f^b <^ c^ d, c ^ {a ^b) ^ d.
No ambiguity exists in these two expressions; each of them definitely
indicates that the synthesis a ^ 6 is to be made first, then a synthesis with
c, and then a S3mthesis of this result with d. Similarly each of the two
expressions d '^ (a '^ 6 /> c), and d ^ {c '^ (a /> b)} indicates unambiguously the
same sequence of operations, though in the final synthesis of d with the
result of the previous syntheses the alternative order of synthesis is adopted
to that adopted in the two previous examples.
14, 15] PRINCIPLES OF ADDITION. 21
But consider the expressions
(a ob) o(c ^ d) and (c ^ d) '^ (a '^ b).
Here the two syntheses a ^b and c ^ d are directed to be made and then
the resulting terms to be combined together. Accordingly there is an
ambiguity as to the sequence in which these sjoitheses a ^^b, c ^d are to
be performed. It has been defined however that a ^b and c ^ d are always
to be unambiguous and mean the same thing. This definition means that
the synthesis ^ depends on no previous history and no varying part of the
environment. Accordingly a ^ 6 is independent ot c^d and these operations
may take place in any sequence 6f time.
The preceding definitions can be connected with the idea of a manifold.
All equivalent things must represent the same element of the manifold.
The synthesis a ^ 6 is a definite unambiguous union which by hypothesis
it is always possible to construct with any two things representing any two
elements of the manifold. This synthesis, when constructed and represented
by its result, represents some third element of the manifold. It is also
often convenient to express this fact by saying that a ^^b represents a
relation between two elements of the manifold by which a third element
of the manifold is generated ; or that the term a '^ 6 represents an element of
the manifold. An element may be named after a term whicb represents it :
thus the element x is the element represented by the term x. The same
element might also be named after any term equivalent to x.
It is obvious that any synthesis of the two terms a and b may be
conceived as an operation performed on one of them with the help of the
other. Accordingly it is a mere change of language without any alteration
of real meaning, if we sometimes consider a ^ 6 as representing an operation
performed on b or on a.
16. Addition. Conceive now that this synthesis which has been defined
above is such that it follows the Commutative and Associative Laws.
The Commutative Law asserts that
a ^b = b ^a.
Hence the two possible orders of synthesis produce equivalent results.
It is to be carefully noticed that it would be erroneous to state the
commutative law in the form that, order is not involved in the synthesis a^b.
For if order is not predicable of the synthesis, then the equation, a /> 6 = 6 /> a,
must be a proposition which makes no assertion at alL Accordingly it is
essential to the importance of the commutative law that order should be
involved in the synthesis, but that it should be indifferent as £Bur as equi-
valence is concerned.
The Associative Law is symbolized by
a ^ b ^ c = a rs (b ^ c)]
where a ^ 6 '^ c is defined in § 14.
1
22 PRINCIPLES OF UNIVERSAL ALGEBRA. [cHAP. II f.
The two laws combined give the property that the element of the
manifold identified by three given terms in successive synthesis is
independent of the order in which the three terms are chosen for the
operation, and also of the internal oixler of each synthesis.
Let a synthesis with the above properties be termed addition; and let
the manifold of the corresponding type be called an algebraic manifold ; and
let a scheme of things representing an algebraic manifold be called an
algebraic scheme. Let addition be denoted by the sign +. Accordingly it
is to be understood that the symbol a + 6 represents a synthesis in which
the above assumptions are satisfied.
The properties of this operation will not be found to vary seriously in the
different algebras. The great distinction between these properties turns
on the meaning assigned to the addition of a term to itself Ordinary
algebra and most special algebras distinguish between a and a + a^ But the
algebra of Symbolic Logic identifies a and a+a. The consequences of these
assumptions will be discussed subsequently.
16. Principles of Subtraction. Let a and b be terms representing
any two given elements of an algebraic manifold. Let us propose the problem,
to find an element w of the manifold such that
x + b = a^
There may be no general solution to this problem, where a and b are
connected by no special conditions. Also when there is one solution, there
may be more than one solution. It is for instance easy to see that in an
algebra which identifies a and a'\' a, there will be at least two solutions if
there be one. For if a? be one answer, then a? + 6 = a? + 6+6 = a. Hence
x + b is another answer.
If there be a solution of the above equation, let it be written in the form,
a N^ 6. Then it is assumed that a^ b represents an element of the mani-
fold, though it may be ambiguous in its signification.
The definition of a^ b is
a ^ b + b^a : (1).
If c be another element of the manifold let us assume that (a^b)^ c
symbolizes the solution of a double problem which has as its solution or
solutions one or more elements of the manifold.
Then av^ 6v^ c + (6 + c)=:a ^ 6v/c + (c + 6)
It follows that the problem proposed by the symbol a^(b + c) has one
or more solutions, and that the solutions to the problem a v^ 6 ^^ c are included
in them.
10] PRINCIPLES OF SUBTKACTION. 23
Conversely suppose that the problem a ^ (6 + c) is solved by one or more
elements of the manifold.
Then by hypothesis a ^ (6 -f- c) + (6 + c) == a ; and hence
{a v^ (6 + c)} + c + 6 = a w (6 4- c) + (t + c) = a.
But if d + c + 6 = a, then ci + c is one value of a v^ 6 and d is one value of
Accordingly a^b^ c is a problem which by hypothesis must have one
or more solutions, and the solutions to a v^ (6 + c) are included in them.
Hence since the solutions of each are included in those of the other, the
two problems must have the same solutions. Therefore whatever particular
meaning (in the choice of ambiguities) we assign to one may also be assigned
to the other. We may therefore write
av^(6 + c) = av/6 ^c (2).
Again we have
a v^ (6 + c) = a v^ (c + 6).
Hence from equation (2),
a'^b^ c = a^ c^h (3).
It may be noted as a consequence of equations (2) and (3), that if
a v^ (6 + c) admit of solutions, then also both a^b and a v^ c admit of
solutions.
Hence lia^b and 6 v^ c admit of solutions ; then aw6 = av/(6v'C + c);
and it follows from the above note that a^ {b^ c) admits of a solution.
Also in this case
av/6-|-c = aw(6wc + c) + c = av/(6v/c)v^c + c, .
from equation (2).
Hence a ^ 6 + c = a v^ (6 ^ c) (4).
We cannot prove that a^b-^-c^a-^c^b, and that a + (6 ^^ c) = a + 1 ^^ c,
without making the assumption that a v^ 6, if it exists, is unambiguous.
Summing up : for three terms a, b and c there are four equivalent forms
symbolized by
(a N^ 6) ^ c = (a ^^ c) v^ 6 = a ^ (6 4- c) = a ^ (c + 6) :
also there are three sets of forms, the forms in each set being equivalent but
not so forms taken from different sets, namely
(a>^ 6) + c = a v^(6 ^ c) = c + (av^6) (a),
{c^ 6) + a = c ^ {b^ a)^a + {c^ b) (^8),
(a + c)>^ 6=(c + a) v^ 6 (7).
Subtraction. Let us now make the further assumption that the reverse
anal}rtical process is unambiguous, that is to say that only one element of
24 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. lU.
the manifold is represented by a symbol of the type a^b. Let us replace in
this case the sign ^ by — , and call the process subtraction.
Now at least one of the solutions of a + b^b is a. Hence in subtraction
the solution of o + 1 — 6 is a, or symbolically a + 6 — J = a. But by definition,
a — 6 + 6 = a.
Hence, a -f-6 — 6 = a — b + b = a (5).
We may note that the definition, a — b + b = a, assumes that the question
a — b has an answer. But equation (5) proves that a manifold may always
without any logical contradiction be assumed to exist in which the subtractive
question a— 6 has an answer independently of any condition between a and 6,
For fix)m the definition, a — b + b, where a — 6 is assumed to have an answer,
can then be transformed into the equivalent form a + 6 — 6, which is a question
capable of an answer without any condition between a and 6. But it may
happen that in special interpretations of an algebra a — 6, though unam-
biguous, has no solution unless a and b satisfy certain conditions. The
remarks of § 7 apply here.
Again a + 6 — c = a + (6 — c + c) — c
= a + (6 — c) + c — c
= a+(6-c) (6).
17. The Null Element. On the assumption that to any question
of the type a — b can be assigned an answer, some meaning must be assigned
to the term a — a.
Now if c be any other term,
c + a — a^c^c + b — b.
Hence it may be assumed that
a — a = 6 — 6.
Thus we may put
a-a = 0 (7);
where 0 represents an element of the manifold independent of a. Let the
element 0 be called the null element. The fundamental property of the null-
element is that the addition of this element and any other element a of the
manifold yields the same element a. It would be wrong to think of 0 as
I necessarily symbolizing mere nonentity. For in that case, since there can be
no differences in nonentities, its equivalent forms a — a and b — b must be not
only equivalent, but absolutely identical ; whereas they are palpably different.
Let any term, such as a - a, which represents the null element be called a
null term.
The fundamental property of 0 is,
a + 0 = a (8).
17 — 19] THE NULL ELEMENT. 25
Other properties of 0 which can be derived from this by the help of the
previous equations are,
0 + 0 = 0;
and a — 0 = a — (6— 6) = a — 6 + 6=a.
Again forms such as 0 — a may have a meaning and be represented by
definite elements of the manifold.
The fundamental properties of 0 — a are symbolized by
6 + (0-a) = 6+0-a = 6-a,
and 6 — (0 — a) = 6 — 0 + tt = 6+a.
Since in combination with any other element the null element 0 dis-
appears, the symbolism may be rendered more convenient by writing — a for
0 — a. Thus — a is to symbolize the element 0 — a.
18. Steps. We notice that, since a = 0 + a, we may in a similar way
consider a or + a as a degenerate form of 0 + a. From this point of view
every element of the manifold is defined by reference to its relation with the
null element. This relation with the null element may be called the step
which leads from the null element to the other element. And by fostening
the attention rather on the method of reaching the final element than on the
element itself when reached, we may call the symbol + a the symbol of the
step by which the element a of the manifold is reached.
This idea may be extended to other elements besides the null element.
For we may write 6 ==: a + (6 — a) ; and 6 — a may be conceived as the »fep
from a to &. The word step has been used* to imply among other things a
quantity ; but as defined here there is no necessary implication of quantity.
The step + a is simply the process by which any term p is transformed into
the term p + a. The two steps + a and — a may be conceived as exactly
opposed in the sense that their successive application starting from any
term p leads back to that term, thus p + a — a=p. In relation to +a,
the step — a will be called a negative step ; and in relation to — a, the
step + a will be called a positive step. The frmdamental properties of steps
are (1) that they can be taken in any order, which is the commutative law,
and (2) that any number of successive steps may be replaced by one definite
tesultant step, which is the associative law.
The introduction of the symbols + a and — a involves the equations
+ (+ a) = + (0 + a) = 0 + a = + a = a,
-(+a) = -(0 + a) = -0-a = -a,
+ (-.a) = + (0-a) = 0-a = -a, '' ^ ^'
— (- a) = — (0 — a) = — 0 + a = + a = a..
19. Multiplication. A new mode of synthesis, multiplication, is now
to be introduced which does not, like addition, necessarily concern terms of a
* Of. Clifford, Elements of Dynumic,
26 _ PRINCIPLES OF UNiyERSAL ALUEBRA. [CHAP. lil,
single algebraic scheme (cf. § 15), nor does it necessarily reproduce as its
result a member of one of the algebraic schemes to which the terms S3nithe-
SATt I sized belong. Again, the commutative and associated laws do not necessarily
hold for multiplication ; but a new law, the distributive law, which defines
the relation of multiplication to addition holds. Any mode of synthesis for
which this relation to addition holds is here called multiplication. The result
of multiplication like that of addition is unambiguous.
Consider two algebraic manifolds; call them the manifolds A and B.
Let a, a', a" etc., be terms denoting the various elements oi A, and let 6, b\ V
etc., denote the various elements of B. Assume that a mode of synthesis is
possible between any two terms, one from each manifold. Let this synthesis
result in some third thing, which is the definite unambiguous product under
all circumstances of this special synthesis between those two elements.
Also let the idea of order between the two things be attributable to their
union in this synthesis. Thus if a and 6 be the two terms of which the
synthesis is being discussed, an order as between a first or h first can be
attributed to this synthesis. Also let only two possible alternative orders
as between a and h exist.
Let this mode of synthesis be, for the moment, expressed by the sign i=^ .
Thus between two terms a, h from the respective manifolds can be generated
the two things a^h and 6 j=: a.
All the things thus generated may be represented by the elements of a
third manifold, call it (7. Also let the symbols a^h and h^a conceived as
representing such things be called terms. Now assume that the manifold
(7 is an algebraic manifold, according to the definition given above (§ 15).
Then its corresponding terms are capable of addition. And we may write
(a i=j 6) -f- (6' j=: a") + etc. ; forming thereby another term representing an ele-
ment of the manifold (7.
The diefinition of the algebraic nature of G does not exclude the potssi-
bility that elements of G exist which cannot be foimed by this synthesis of
two elements from A and B respectively. For (a:=:6) + (6"^=^ d) is by definition
an element of G ; but it vrill appear that this element cannot in general
be formed by a single synthesis of either of the types a^^^ ^ h^^^ or ¥^^ ^ a^^K
Again a + a' + a'' + etc., represents an element of the manifold .^,and
6 + 6' + 6" -f- etc., represents an element of the manifold B, Hence there are
elements of the manifold G represented by terms of the form
(a-f-a'-f- a" + etc.);=:(6 + 6' + 6" + ...),
and (6 + 6' + 6" + etc.)^(a + a 4-a"H- ...)■
Now let this synthesis be termed Multiplicdtion, when such expreS'
sions as the above follow the distributive, law as defined by equations (10)
below.
For multiplication let the synthesis be denoted by x or by mere juxta-
20] MULTIPLICATION. 27
position. Then the definition of multiplication yields the following symbolic
statements
(a-\-a^)b = ab+a% \
6(a + a') = 6a + 6a', ^ ^^^^•
(b + b')a = ba'^Va,
It will be noticed that the general definition of multiplication does not
involve the associative or the commutative law.
20. Orders of Algebraic Manifolds. Consider a single algebraic
manifold A, such that its elements can be multiplied together. Call such a
manifold a self-multiplicative manifold of the first order. Now the products
of the elements, namely cut, aa\ a'a, etc., by hypothesis form another alge-
braic manifold ; call it B, Then B will be defined to be a manifold of the
second order.
Now let the elements of A and B be capable of multiplication, thus
forming another algebraic manifold C, Let C be defined to be a manifold
of the third order. Also in the same way the elements of A and C form
by multiplication an algebraic manifold, D, of the fourth order ; and so on.
Further let the elements of any two of these manifolds be capable of
multiplication, and each manifold be self-multiplicative.
Let the following law hold, which we may call the associative law for
manifolds. The elements formed by multiplying elements of the manifold
of the mih order with elements of the manifold of the nth order belong to
the manifold of the (m + n)th order.
• Thus the complete manifold of the mth oi-der is formed by the multiplica-
tion of the elements of any two manifolds, of which the sum of the ordere
forms m, and also by the elements deduced by the addition of elements
thus formed.
For instance <xa, aaW\ aa'a'W, represent elements of the manifolds of
the second, third, and fourth oi-ders respectively; also cut represents an element
of the manifold of the second order. Also a" (oaf) is an element of the mani-
fold of the third order ; and (oa') (a'V) is an element of the manifold of the
fourth order; and aa'(aaW") is an element of the manifold of the sixth order;
and so on.
Such a system of manifolds will be called a complete algebraic system.
In special algebras it will be found that the manifold of some order,
say the mth, is identical with the manifold of the first order. Then the
manifold of the m 4- 1th order is identical with that of the second order, and
so on.
Such an algebra will be said to be of the m - 1th species. In an algebra
of the first species only the manifold of the first order can occur. Such
28 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. III.
an algebra is called linear. The Calculus of Exteusion, which is a special
algebra invented by Grassmann, can be of any species.
It will save symbols, where no confusion results, to use dots instead
of brackets. Thus a" {(w!) is written a'. aa\ and {aa') {a"d") is written
aa . a'V, and so on. A dot will be conceived as standing for two opposed
bracket signs, thus )(, the other ends of the two brackets being either other
dots or the end or beginning of the row of letters. Thus ah . cd stands for
{ab) {cd)y and is not (a6) cd, unless in the special algebra considered, the two
expressions happen to be identical ; also ab . cde ,fg stands for {ab) (cde) (fg).
It will be noticed that in these examples each dot has been replaced by two
opposed bracket signs. An ingenious use of dots has been proposed by Mr
W. E. Johnson which entirely obviates the necessity for the use of brackets.
Thus a {b (cd)] is written a.,b .cdy and a [b {c (de)]] is written a,..b..c,de.
The principle of the method is that those multiplications indicated by the
fewest dots are the first performed. Thus a {b{cd)} (ef) is written a,.b.cd,.ef,
and a {b {cd)] ef is written a.,b,cd.,e ...f, where in the case of equal numbers
of dots the left-hand multiplication is first performed.
21. The Null Element. Returning to the original general conception
of two algebraic manifolds A and B of which the elements can be multiplied
together, and thus form a third algebraic manifold C; let Oi be the null
element of -4, Oa the null element of B, and 0, the null element of C,
Then if a and b represent any two elements of the manifolds A and B
respectively, we have
a + Oi = a, and 6 + 0, = 6.
Hence (a + Oi) 6 = aft = ab + Oib.
Accordingly, Oib = O3 .
Similarly, 6O1 = 0, = aOj = 0^.
No confusion can arise if we use the same symbol 0 for the null elements
of each of the three manifolds.
Accordingly, 0a = aO = O6 = 6O = O (11).
It will be observed that a null element has not as yet been defined for
the algebraic manifold in general ; but only for those which allow of the
process of subtraction, as defined in § 16. Thus manifolds for which the
relation a + a^a holds are excluded from the definition.
In order to include these manifolds let now the null element be defined
as that single definite element, if it exist, of the manifold for which the
equation
a + 0 = a,
holds, where a is a/ny element of the manifold.
It will be noted that for the definite element a the same property may
21, 22] THE NULL ELEMENT. 29
hold for a as well as for 0; since in some algebras a-\-a = a. But 0 is
defined to be the single element which retains this property with all
elements. Then in the case of multiplication equations (11) hold.
22. Classification of Special Algebras. The succeeding books of
this work vdll be devoted to the discussion and compaiison of the leading
special algebraa It remains now to explain the plan on which this in-
vestigation will be conducted.
It follows from a consideration of the ideas expounded in Chapter i. that
it is desirable to conduct the investigation of a calculus strictly in connection
with its interpretations, and that without some such interpretation, however
general, no^ great progress is likely to be made. Therefore each special
algebra will, as far as possible, be interpreted concurrently with its in-
vestigation. The interpretation chosen, where many are available, will
be that which is at once most simple and most general ; but the remaining
applications will also be mentioned with more or less fulness according as
they aid in the development of the calculus. It must be remembered,
however, in explanation of certain obvious gaps that the investigation is
primarily for the sake of the algebra and not of the interpretation.
No investigation of ordinary algebra will be attempted. This calculus
stands by itself in the fundamental importance of the theory of quantity
which forms its interpretation. Its formulae will of course be assumed ^
throughout when required.
In the classification of the special algebras the two genera of addition
form the first ground for distinction.
For the purpose of our immediate discussion it will be convenient to
call the two genera of algebras thus formed the non-numerical genus and
the numerical genus.
In the non-numerical genus investigated in Book II. the two symbols
a and a'{'a, where a represents any element of the algebraic manifold,
are equivalent, thus a = a + a. This definition leads to the simplest and
most rudimentary type of algebraic symbolism. No symbols representing
number or quantity are required in it. The interpretation of such an algebra
may be expected therefore to lead to an equally simple and fundamental
science. It will be found that the only species of this genus which at present
has been developed is the Algebra of Symbolic Logic, though there seems no
reason why other algebras of this genus should not be developed to receive
interpretations in fields of science where strict demonstrative reasoning with-
out relation to number and quantity is required. The Algebra of Symbolic
Logic is the simplest possible species of its genus and has accordingly the
simplest interpretation in the field of deductive logic. It is however always
desirable while developing the symbolism of a calculus to reduce the inter-
pretation to the utmost simplicity consistent with complete generality.
30 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. IIL
Accordingly in discussing the main theory of this algebra the difficulties
peculiar to Symbolic Logic will be avoided by adopting the equally general
interpretation which considers merely the intersection or non-intersection
of regions of space. This interpretation will be developed concurrently with
the algebra. After the main theory of the algebra has been developed, the
more abstract interpretation of Symbolic Logic will be introduced.
In the numerical genus the two symbols a and a + a are not equivalent.
The symbol a + a is shortened into 2a ; and by generalization of this process
a symbol of the form fa is created, where f is an ordinary algebraical
quantity, real or imaginary. Hence the general type of addition for this
genus is symbolized by f a + 176 + 5<^ + etc., where a, b, c, etc. are elements
of the algebraic manifold, and f, rf, ^, etc. are any ordinary algebraic
quantities (such quantities being always symbolized by Greek letters, c£
Book III. Chapter I. below). There are many species of algebra with im-
portant interpretations belonging to this genus; and an important general
theory, that of Linear Associative Algebras, connecting and comparing an
indefinitely large group of algebras belonging to this genus.
The special manifolds, which respectively form the interpretation of all
the special algebras of this genus, have all common properties in that
they all admit of a process symbolized by addition of the numerical
type. Any manifold with these properties will be called a 'Positional
Manifold.' It is therefore necessary in developing the complete theory of
Universal Algebra to enter into an investigation of the general properties
of a positional manifold, that is, of the properties of the general type of
numerical addition. It will be found that the idea of a positional manifold
will be made more simple and concrete without any loss of generality
by identifying it with the general idea of space of any arbitrarily assigned
number of dimensions, but excluding all metrical spatial ideas. In the
discussion of the general properties of numerical addition this therefore will
be the interpretation adopted as being at once the most simple and the
most general. All the properties thus deduced i^ust necessarily hold ^or
any special algebra of the genus, though the scale of the relative importance
of different properties may vary in different algebras. Positional manifolds
are investigated in Book III.
Multiplication in algebras of the numerical genus of course follows all
the general laws investigated in this chapter. There is also one other
general law which holds throughout this genus. The product of {a and 17&
({ and 17 being numbers) is defined to be equivalent to the product of ^
(ordinary multiplication) into the product of a and b. Thus in symbols
^.7fb = (rfoh, tfb. ^^ ^ba ;
where the juxtaposition of f and 17 always means that they are to be
multiplied according to the ordinary law of multiplication for numbers.
If this law be combined with equation 10 of § 19, the following general
22] CLASSIFICATION OF SPECIAL ALGEBRAS. 31
equation must hold: let 61, eg, ... e, be elements of the manifold, and let
Greek letters denote numbers (i.e. ordinary algebraic quantities, real or
imaginary), then
It follows that in the numerical genus of algebras the successive derived
manifolds are also positional manifolds, as well as the manifold of the first
order.
In the classification of the special algebras of this genus the nature of the
process of nmltiplication as it exists in each special algebra is the guide.
The first division must be made between those algebras which involve a
complete algebraical system of more than one manifold and those which
involve only one manifold, that is, between algebras of an order higher
than the first and between linear algebras (cf. § 20). It is indeed possible
to consider all algebras as linear. But this simplification, though it has
very high authority, is, according to the theory expounded in this work,
fallacious. For it involves treating elements for which addition has no mean-
ing as elements of one manifold ; for instance in the Calculus of Extension
it involves treating a point element and a linear element as elements of one
manifold capable of addition, though such addition is necessarily meaningless.
The only known algebra of a species higher than the first is Grassmann's
Calculus of Extension ; that is to say, this is the only algebra for which this
objection to its simplification into a linear algebra holds good. The Calculus
of Extension will accordingly be investigated first among the special algebras
of the numerical genus. It can be of any species (cf. § 20). The general
type of manifold of the first algebraic order in which the algebra finds its
interpretation will be called an Extensive Manifold. Thus an extensive
manifold is also a positional manifold.
In Book IV. the fundamental definitions and formulae of the Calculus
of Extension will be stated and proved. The calculus will also be applied
in this book to an investigation of simple properties of extensive manifolds
which, though deduced by the aid of this calculus, belong equally to the
more general type of positional manifolds. One type of formulae of the
algebra will thus receive investigation. Other types of formulae of the
same algebra are developed in Books V., VI. and VII., each type being
developed in conjunction with its peculiar interpretation. The series of
interpretations will form, as they ought to do, a connected investigation
of the general theory of spatial ideas of which the foundation has been laid
in the discussion of positional manifolds in Book III.
This spatial interpretation, which also applies to the algebra of Symbolic
Logic, will in some form or other apply to every special algebra, in so far as
interpretation is possible. This fact is interesting and deserves investigation.
32 PRINCIPLES OF UNIVERSAL ALGEBRA. [CHAP. III. 22
The result of it is that a treatise on Universal Algebra is also to some
extent a treatise on certain generalized ideas of space.
In order to complete this .subsidiary investigation an appendix on a mode
of arrangement of the axioms of geometry is given at the end of this volume.
The second volume of this work will deal with Linear Algebras. In
addition to the general theory of their classification and comparison, the
special algebras of quaternions and matrices will need detailed development.
Note. The discussions of this chapter are largely based on the ' Ueber-
sicht der allgemeinen Formenlehre* which forms the introductory chapter
to Grassmann's Ausdehnungslehre von 1844.
Other discussions of the same subject are to be found in Hamilton's
Lectures on Quaternions, Preface; in Hankel's Vorlesungen uber Complexe
ZahUn (1867) ; and in De Morgan's Trigonometiy and Double Algebra, also
in a series of four papers by De Morgan, * On the Foundation of Algebra/
Transactions of the Cambridge Philosophical Society, vols. vii. and VIIL,
(1839. 1841, 1843, 1844).
BOOK II.
THE ALGEBRA OF SYMBOLIC LOGIC.
w.
CHAPTER 1.
The Algebra of Symbolic Logic.
23. Formal Laws. The Algebra of Symbolic Logic* is the only known
member of the non-numerical genus of IJDiversal Algebra (cf. Bk. I., Ch. ill.,
§ 22).
It will be convenient to collect the formal laws which define this special
algebra before considering the interpretations which can be assigned to the
symbols. The algebra is a linear algebra (cf. § 20), so that all the terms used
belong to the same algebraic scheme and are capable of addition.
Let a, b, c, etc. be terms representing elements of the algebraic manifold
of this algebra. Then the following symbolic laws hold.
(1) The general laws of addition (cf. Bk. I. Ch. in., g 14, 15) :
a + 6 = 6 -f a,
a + 6 + c = (a + 6) + c = a + (6 + c).
(2) The special law of addition (cf § 22) :
a^ a = a,
(3) The definition of the null element (cf. § 21) :
a + 0 = a.
(4) The general laws of multiplication (cf § 19) :
c(a-\'b)^ca + cb,
(a-\-b)c^ac + be,
(5) The special laws of multiplication :
ab=^ba,
ahc = ab.c=^a.bc,
* This algebra in all essential partioulars was invented and perfected bj Boole, cf. his work
entitled, An Investigation of the Laws of Thought^ London, 1854.
3—2
36 THE ALGEBRA OF STMBOUC LOGIC. [CHAP. 1.
(6) The law of ' absorption ' :
a + a& = a.
This law includes the special law (2) of addition.
(7) The definition of the ' Universe.* This is a special element of the
manifold, which will be always denoted in future by i, with the following
property :
ai = a,
(8) Supplementary elements. An element b will be called supple-
mentary to an element a if both a + 6 = t, and a6 = 0. It will be proved that
only one element supplementary to a given element can exist ; and it will be
assumed that one such element always does exist. If a denote the given
element, a will denote the supplementary element. Then a will be called
the supplement of a. The supplement of an expression in a bracket, such
as (a + 6), will be denoted by " (a + 6).
The theorem that any element a has only one supplement follows from
the succeeding fundamental proposition which develops a method of pi'oof of
the equivalence of two terms.
Proposition I. If the equations xy = xz, and a? -f y = a? + ^, hold simul-
taneously, then y = z.
Multiply the second equation by x, where x is one of the supplements of
X which by hypothesis exists.
Then ^ (a? + y) = ac (a? + z).
Hence by (4) xx -{-xy = xx -hxz,
hence by (8) and (3) xy = xz.
Add this to the first equation, then by (4)
{x-hx)y=={x + x)z,
hence by (8) iy =-t5,
hence by (7) y = z.
Corollary I. There is only one supplement of any element x. For if
possible let x and x' be two supplements of x.
Then a^ = 0 = xx\ and a? + al = i = a? -f a?'.
Hence by the proposition, x « x\
Corollary II. I{x = y, then S = y.
Corollary III. i; = 0, and 6 = i.
Corollary IV. i = a? ;
where x means the supplement of the supplement of x. The prooiEs of these
corollaries can be left to the reader.
24] FORMAL LAWS. 37
Proposition II. (a? + y) (a? + ^r) as a; + yz.
For (x •\' y){x -^^ z) ^ XX •\' xy -\- xz •\' yz — X -^^ a:{y -{• z) •\' yz
^x-\-yz, by (6).
Proposition III. a^O = 0 = Oa?, and a? + i = i = i + a;.
The first is proved in § 21. The proof of the second follows at once
from (6) and (7).
24. Reciprocity between Addition and Multiplication. A reci-
procity between addition and multiplication obtains throughout this algebra ;
so that corresponding to every proposition respecting the addition and
multiplication of terms there is another proposition respecting the multi-
plication and addition of terms. The discovery of this reciprocity was first
made by C. S. Peirce*; and later independently by Schroder f.
The mutual relations between addition and multiplication will be more
easily understood if we employ the sign x to represent multiplication. The
definitions and fundamental propositions of this calculus can now be arranged
thus.
The Conmiutative Laws are (cf. (1) and (5))
xxy = yxx,) ^ ^'
The Distributive Laws are (cf. (4) and Prop. II.)
X x(y-h z) ^{x X y) -^ (x X z),) ^
X -h {y X z)=^ (x -\- y) X (x + z),) ^ ^'
The Associative Laws are (cf. (1) and (5))
x + (y + z)^x-hy + z,] ^ .^
X x(ifx z) = xxy xz.) ■ ^'
The Laws of Absorption are (cf. (6))
x-\'{xxy) = x^x-\-x,) .j^.
a?x(a:-hy)= a? = a?xa?.J
The properties of the Null element and of the Universe are (cf. (3), (7),
and Prop. III.),
::oro] ■ <«■
a? + 0 = a?,l
a? X i = X,)
in
The definition of the supplement of a term gives (cf (8) and Prop. I.)
(G).
x-\- x = %,)
a; X « = 0. 1
* Proe, of the American Academy of Arts and Seiencei, 1867.
t Der Operatiomkreit des Lo0ikkalk(iU, 1877.
S8 THE ALGEBRA OF SYMBOUC LOGIC. [CHAP. 1.
There can therefore be no distinction in properties between addition and
multiplication. All propositions in this calculus are necessarily divisible into
pairs of reciprocal propositions; and given one proposition the reciprocal
proposition can be immediately deduced from it by interchanging the signs
+ and X , and the terms % and 0. An independent proof can of course
always be found : it will in general be left to the reader.
Also any interpretation of which the calculus admits can always be
inverted so that the interpretation of addition is assigned to multiplication,
and that of multiplication to addition, also that of t to 0 and that of 0 to i,
25. Interpretation. It is desirable before developing the algebraic
formulae to possess a simple and general form of interpretation (cf. § 7 and
§22>
Let the elements of this algebraic manifold be regions in space, each
region not being necessarily a continuous portion of space. Let any term
symbolize the mental act of determining and apprehending the region which
it represents. Terms are equivalent when they place the same region before
the mind for apprehension.
Let the operation of addition be conceived as the act of apprehending in
the mind the complete region which comprises and is foimed by all the
regions represented by the terms added. Thus in addition the symbols
represent firstly the act of the mind in apprehending the component regions
represented by the added terms and then its act in apprehending the
complete region. This last act of apprehension determines the region which
the resultant term represents. This interpretation of terms and of addition
both satisfies and requires the formal laws (1) and (2) of § 23. For the
complete region does not depend on the order of apprehension of the com-
ponent regions ; nor does it depend on the formation of subsidiary complete
regions out of a selection of the added terms. Hence the commutative and
associative laws of addition are required. The law, a + a=a, is satisfied
since a region is in no sense reduplicated by being placed before the mind
repeatedly for apprehension. The complete region represented by a + a re-
mains the region represented by a. This is called the Law of Unity by
Jevons (cf. Pure Logic, ch. vi).
The null element must be interpreted as denoting the non-existence of
a region. Thus if a term represent the null element, it symbolizes that
the mind after apprehending the component regions (if there be such)
symbolized by the term, further apprehends that the region placed by the
term before the mind for apprehension does not exist. It may be noted that
the addition of terms which are not null cannot result in a null term. A
null teim can however arise in the multiplication of terms which are not null.
Let the multiplication of terms result in a term which represents the
entire region common to the terms multiplied. Thus xyz represents the
25, 26] INtERPBEtATIOK. 39
entire region which is at once incident in the regions x and y and z. Hence
the term xy symbolizes the mental acts first of apprehending the regions
symbolized by x and y, and then of apprehending the region which is their
complete intersection. This final act of apprehension determines the region
which «y represents
This interpretation of multiplication both satisfies and requires the
distributive law, numbered (4) in § 23, and the commutative and associative
laws marked (5) in § 23. The law, aa = a, which also occurs in (5) of § 23 is
satisfied ; for the region which is the complete intersection of the region a
with itself is again the region a. This is called the Law of Simplicity by
Jevons (cf. loc. cit).
The Law of Absorption (cf (6) § 23) is also required and satisfied. For
the complete region both formed by and comprising the regions a and ah is
the region a, and the final act of apprehension symbolized by a + a6 is that of
the region a. Hence
a 4- aft = a.
This interpretation also requires that i( x-^y^x, then y^xy. And this
proposition can be shown to follow from the formal laws (cf § 26, Prop. viii.).
The element called the Universe (cf § 23 (7)), must be identified with all
space ; or if discourse is limited to an assigned portion of space which is to
comprise all the regions mentioned, then tlie Universe is to be that complete
region of space.
The term supplementary (cf § 23 (8)) to any term a represents that
region which includes all the Universe with the exception of the region a.
The two regions together make up the Universe ; but they do not overlap, so
that their region of intersection is non-existent.
It follows that the supplement of the Universe is a non-existent region,
and that the supplement of a non-existent region is the Universe (cf Prop. I.
Cor. 3).
26. Elementary Propositions. The following propositions of which
the interpretation is obvious can be deduced from the formal laws and from
the propositions already stated.
Proposition IV. If a? -h y = 0, then a? = 0, y = 0.
For multipljdng by x, x{x-\-y)^ 0.
But a;(ar + y) = a?, by (6) § 23.
Hence a? = 0. Similarly, y = 0.
The reciprocal theorem is, if ajy = i, then a? = i, y = i.
Proposition V. x + y = x + yx, and xy^x{y-\-x).
For X + y ^ X -\' y {x -hx) = X + yx + yx ^ X + yx.
The second part is the reciprocal proposition to the first part.
40 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. I.
Proposition VI. - {xy) = x+y,
and ■■(fl? + y) = «y-
For by Prop, v., x-^y-x + xy^x-^-xy.
Hence ay +(x + y) = xy'hay + x=^x(if + y)+x
•*• X<^+^f i^^ =flj + ^ = i. — — v)
Also {vy(x-\-y)-(cxy + xyy = 0. - - — --(?-)
ZtnUa^A^^iS) o-^z.) . Hence by § 23 (8) x + y = - (xy).
The second part is the reciprocal of the first part. Also it can be deduced
from the first part thus :
-(xy) = x + y = x + y.
Taking the supplements of both sides
"(^ + y) = "(^)=^.
CoROiJ^ARY. The supplement of any complex expression is found by
replacing each component term by its supplement and by interchanging
+ and X throughout.
Proposition VII. If xy = xz, then
osy=:xz, and x+y^x-\-z.
For taking the supplement of both sides of the given equation, by
Prop. VI.,
x-^-y^x + z.
Multiplying by a?, xiy = xz.
Again taking the supplement of this equation, then
x + y =^x + z.
The reciprocal proposition is, if x-\-y = x + z, then x + y = x + z, and
xy = xz.
Proposition VIII. The following equations are equivalent, so that from
any one the remainder can be derived :
y = xy, x-\-y = x, xy-0, xi-y^i.
Firstly : assume y = xy.
Then X'\-y^x + xy — x,
And xy =^ xxy = 0.
And a? + j^ = a? + " (xy) = a? + ^ + y = i.
Secondly: assume x + y=ix.
m
Then «y=(«H"y)y *y.
Hence the other two equations can be derived as in the first case.
Thirdly: assume ^ = 0.
1
i
27] ELEMENTARY PROPOSITIONS. 41
Then y^(x-\-x)y^0Dy ^-xy^xy.
Hence the other two equations can be derived.
Fourthly: assume x-\-y = i.
Then taking the supplements of both sides
Hence by the third case the other equations are true.
Corollary. By taking the supplements of the first and second equations
two other forms equivalent to the preceding can be derived, namely
y^x-hy, ^ = x.
Proposition IX. If x^xyz, then x^xy = xz, and if x = x-{-y-\' z,
then x = x + y = x + z.
For osy^xy{z-{-z)=^ xyz + xyz = a? + xyz = x, from (6) § 23.
The second part of the proposition is the reciprocal theorem to the first ^
part. ('X^^4-i)(T^^4i) ^7-^^ ^ ii-\rX(H^iy- X Pn^.JL c^< ^1 f^IJ^')-
Corollary. A similar proof shews that if z=z{xu+yv), then z^z(x'{-y);
and that if ir = 2:+(a? + w)(y-f v), then z = z + xy.
27. Classification. The expression a? + y + £r + , . . , which we can
denote by u for the sake of brevity, is formed by the addition of the
regions x, y, etc. Now these regions may be overlapping regions: we re-
quire to express u as a sum of regions which have no common part. To
this problem there exists the reciprocal problem, given that u stands for
the product xyz.,., to express t^ as a product of regions such that the sum
of any two completes the universe. These problems may be enunciated and
proved sjmibolically as follows.
Proposition X. To express t^ (= a? + y + 2: + . . .), in the form
X+T+Z+...;
where X, F, Z have the property that for any two of them, Y and Z say, the
condition FZ = 0, holds.
Also to express u (= xyz...) in the form XYZ,.. ; where X, F, Z have the
property that for any two of them, F and Z say, the condition F+Z = i,
holds.
Now from Prop. IV., i{ x(y + z)^ 0, then an/ — 0,xz^ 0. Hence for the
first part of the proposition the conditions that Z, F, Z, etc. must satisfy
can be expressed thus,
Z(F+Z + r-|-...)=cO, F(Z+y+...) = 0, Z(r+...) = 0,etc.
Now by Prop. V., u^x-{'y-\-z-\- ...
42 THE ALGEBRA OB* SYMBOLIC LOGIC. [CHAP. L
and y-f ir + ^+... =y + y(2r + ^+ ...);
and z + t + ,..=z +z(t'\' .,,).
Proceeding in this way, we find
u = x-\-xy-^ xyz + xyzt + . . . .
Hence we may write
X^x, Y=xy, Z^xyZy etc.
It is obvious that there is more than one solution of the problem.
Again for the second part of the proposition, consider
S = ^ + y + ^ + . . . .
By the fii-st part of the proposition,
S = iic + a^ + xyz + xyzi +
Here any two terms, 1" and Z, satisfy the condition YZ = 0.
Taking the supplements of these equations,
= xyz. . .
= '-(^'\-xy ^-xyz-^- .„)
Hence we may write XYZ.., for xyz...y where X = a?, Y=x-\-y,
Z^x + y + Zj etc. and any two of X, F, Z, etc., for instance Fand Z, satisfy
the condition F+ Z= i.
It is obvious that there is more than one solution of this problem.
These problems are of some importance in the logical applications of the
algebra.
28. Incident Regions. The symbolic study of regions incident (cf. § 10)
in other regions has some analogies to the theory of inequalities in ordinary
algebra. These relationships also partly possess the properties of algebraic
equations. Two mixed symbols have therefore been adopted to express them,
namely 4 ^^'^ ^ (cf Schroder, Algebra der Logik). Then, y^Xy expresses
that y is incident in x ; and x^ y expresses that x contains y. Expressions
of this kind will be called, borrowing a term from Logic, subsumptions.
Then a subsumption has analogous properties to an inequality. The Theoty
of Symbolic Logic has been deduced by C. S. Peirce from the type of
relation symbolized by ^, cf. American Journal of MaJlk&nuitics, Vols, ill
and Vll (1880, 1885). His investigations are incorporated in Schroder's
Algebra der Logik.
In order to deduce the properties of a subsumption as far as possible
purely symbolically by the methods of this algebra, it is necessary to start
from a proposition connecting subsumptions with equations. Such an initial
proposition must be established by considering the meaning of a subsumption.
28] INCIDENT REGIONS. 43
Proposition XI. liy^x, then y^xy\ and conversely.
For if y be incident in x, then y and ayy denote the same region.
The converse of this proposition is also obvious.
It is obvious that any one of the equations proved in § 26, Prop. VIII.,
to be equivalent to y = xy is equivalent to y ^x. In fact the subsumptiou
y^x may be considered as the general expression for that relation between
X and y which is implied by any one of the equations of Prop. VIII. It follows
that an equation is a particular case of a subsumption.
Corollary, xz^x^x-k-z.
Proposition XII. If a; :): y, and y^z\
then x^z.
For by Prop. XI. and by § 26, Prop. IX.
z — zy = zxy = zx.
Hence x^z,
^ Proposition XIII. \i x^y, and y^x\ then x = y.
For since ^^V^ then y — xy.
And since y^^> iki^xa y^^x-^-y.
xHence y = xy = x{x + y) = x.
Proposition XIV. U x^y, and u^v\
then iix^vy, and u-hx^v + y.
For y = yx, and v = vu ; hence vy = yxvu = vy . xu.
Therefore ux^vy.
Also a? = a? + y, and u = u + v;
hence a7 + w = iP + y + M + t; = (a:+M)H-(y + v).
Therefore x-j-u^y + v.
Corollary. U x^y, and u — v\ then iix ^ vy, and u-{-x^v-\-y.
For v = vu, and u = w + 1; ; hence the proof can proceed exactly as in the
proposition.
The proofe of the following propositions may be lefb to the reader.
■ Proposition XV. If x^yytheny^x.
Proposition XVI. 1{ z^xy, then z^x, z^y, z^x + y.
Proposition XVII. If z 4 ^y, then xy^z, x + y^z.
Proposition XVIII. Uz^x + y, then z^x, z^y, z^xy.
Proposition XIX. Itz^x + y, then xy^z, x + y^z.
Proposition XX. If xz 4 y, and x^y -^z, then x^y.
44 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. I. 28.
Proposition XXI. It z^wu-^yv, then s^x + y.
The importance of Prop. XXI. demands that its proof be given.
By Prop. IX. Cor., z = z (am + yv) = 5 (a? + y).
Therefore -a 4 ^ + y-
Corollary. If z^xu-^-yv, then z^x-\-y;
that is, xu-k-yv^x-^y.
Prop.* XXII. If z^{x-\- u) (y + v), then z ^ xy.
Corollary, (x -\-u)(y + v)^ xy,
* This proposition, which I had overlooked, was pointed out by Mr W. E. Johnson.
CHAPTER II.
The Algebra of Symbolic Logic {continiLed).
29. Development. (1) The expression for any region whatsoever may
be written in the form ox + Imc] where x represents any region.
For let z be any region. Now x + x = %.
Hence z^^x-^x
^ZX'\'ZX,
Now Xeta^zx-^- van, and b — zx-^ vx, where u and v are restricted by no
conditions.
Then ax + bx^ {zx + vai) x + {zx + vx) x^zx + zx^z.
Hence by properly choosing a and b, ax-^-lw can be made to represent
any region z without imposing any condition on x.
Again the expression for any region can be written in the form
(a + a?) (6 + x),
where x represents any other region. For by multiplication
(a H- a?) (6 + a) = oft -f oS + 6a? = (a + ab) S + (6 H- ah) x = a^ + bx.
This last expression has just been proved to represent the most general
region as &r as its relation to the term x is concerned.
(2) Binomial expressions of the form cuc-^bx have many important
properties which must be studied. It is well to notice at once the follow-
ing transformations :
cur + te = (6 + a?) (a + «) ;
" {ax + Iko) = (a + x) (6 +a?)
(ax -h bx) (ex -h cte) = acx + bdx ;
aa7 + te + c=s(a + c)aj-|-(6 + c)«;
ax'^bx^ax-\'bx + ah.
46 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
(3) Let f{x) stand for any complex expression formed according to the
processes of this algebra by successive multiplications and additions of x and x
and other terms denoting other regions. Then /(a?) denotes some region with
a specified relation to x. But by (1) of this article f{x) can also be written
in the form (ix-\-bx. Furthermore a and h can be regarded as specified by
multiplications and additions of the other terms involved in the formation of
f{x) without mention of x. For if a be a complex expression, it must be
expressible, by a continual use of the distributive law, as a sum of products
of which each product either involves ar or S or neither. Since a only appears
when multiplied by a?, any of these products involving « as a factor can be
rejected, since xx = 0\ also any of these products involving a; as a factor can
be written with the omission of x, since xx = x. Hence a can be written in
a form not containing x or x. Similarly h can be written in such a form.
(4) Boole has shown how to deduce immediately from /(a?) appropriate
forms for a and h. For write /(a?) = ax-\-bx. Let i be substituted for a?, then
f(t) = at + 6i; = ai H- 0 = a.
Again let 0 be substituted for x, then
/(O) = aO + 60 = 0 + W = 6.
Hence /(a?) =r/(») x +/(0) x.
For complicated expressions the rule expressed by this equation shortens
the process of simplification. This process is called by Boole the develop-
ment of/ (a?) with respect to x.
The reciprocity between multiplication and addition gives the reciprocal
rule
/(^) = {/(0) + «){/(t') + S}.
(5) The expressions /(»') and /(O) may involve other letters y, z, etc.
They may be developed in respect to these letters also.
Consider for example the expre8sion/(a:, y, z) involving three letters.
/(<», y, i)=f(i, y, «)a! +/(0, y, z)x,
f(i, y, z)=/(i, t, z)y+fii, Q, g)y,
f(i, i. *)=/(». i, i)z+f{x. i, 0)i,
f(i, 0, z) ^/(i, 0, i) z +f{i, 0, 0) z,
/(O. y, z) =/(0. i. z) y +/((), 0, z) y,
f(0, i, z) =/(0, i, i) z +/(0, i, 0) z.
/(O, 0, z) =/(0, 0, t) z +/(0. 0, 0) z.
Hence by substitution
/(«, y, z) =f{%, i, i) asyz +/(0, i, i) xyz +f(i, 0, t) asyz +/(t, t, 0) xyz
+/(i, 0, 0) xyz +/(0, i, 0) xyz +/(0, 0. t) ^z +/(0, 0, 0) xyz.
30] DEVELOPMENT. 47
The reciprocal fonnula, owing to the brackets necessary, becomes too
complicated to be written down here.
Let any term in the above developed expression for f{x, y, z), say
/(O, i, 0) xyz, be called a constituent term of the type xyz in the develop-
ment.
(6) The rule for the supplement of a binomial expression given in
subsection (1) of this article, namely ~" {ax + bx) = aa? + fe, can be extended
to an expression developed with respect to any number of terms x^y, z, ....
The extended rule is that if
/(a?, y, Zy ..,)^axyz... + ... H-gf^^...,
then ""/(^> y> ^> ...) = dxyz .., + ... +^^^....
In applying this proposition any absent constituent term must be
replaced with 0 as its coefficient and any constituent term with the form
xyz... must be written ixyz.., so that % is its coefficient.
For assume that the rule is true for n terms x, y, z... and let ^ be an
(n + l)th term.
Then developing with respect to the n terms a?, y, j&, . . .
f(^,y*^, ...t)-Axyz.:.^',..-\-Owgz...y
where the products such as xyz ..., ..., xyz ... do not involve t, and
Then the letters a, a\ ..., g, g' are the coefficients of constituent terms
of the expression as developed with respect to the n + 1 terms Xy y, z,,,,t.
Hence by the assumption
"/(a?, y, Zy.,.t)-Axyz .,. + ... + G«p....
But by the rule already proved for one term,
A -dt-^ctiy ..., 0=gt-\-gt.
Hence the rule holds for {n + l)th terms. But the rule has been proved
for one term. Thus it is true always.
30. Elimination. (1) The object of elimination may be stated thus :
Given an equation or a subsumption involving certain terms among others,
to find what equations or subsumptions can be deduced not involving those
terms.
The leading propositions in elimination are Propositions XXI. and XXII.
of the last chapter, namely that, if z^xu-^yv, then z^x-j-y; and if
jer :^ (a? + w) (y + v), then z^xy; and their Corollaries that, xu-^yv^x + y, and,
(a: + w)(yH-v)a^ajy.
(2) To prove that if aa? + te = c, then a + b^c^ah.
Eliminating x and x by the above-mentioned proposition from the
equation
c=*aa?-»-6«, c^a + fc.
48 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP, XL
Also takiug the supplementary equation,
Hence from above
c^a + b.
Taking the supplementary subsumption,
oft = *" (a + 6)
Therefore finally
a + b^c^ab.
The second part can also be proved* by taking the reciprocal equation,
c = (tt •+ ^) (6 + x), and by using Prop. XXII. Corollary.
The same subsumptions, written in the supplementary form, are
a + 6 a^ c :^ ofc.
(3) By Prop. XI. each of these subsumptions is equivalent to an equation,
which by Prop. VIII. can be put into many equivalent forms.
Thus a + 6 s^ c, can be written
c = c(a-\'b)',
and this is equivalent to
a6c = 0.
And c^ab, can be written
ab = abc;
and this is equivalent to abc = 0.
(4) Conversely, if
c^^a-^b,
^ab
f6,)
> J
then it has to be shown that we can write dx + bx^c; where we have to
determine the conditions that x must fulfil. This problem amounts to
proving that the equation dx + b^^c has a solution, when the requisite
conditions between a, b, c are fulfilled. The solution of the problem is
given in the next article (cf. § 31 (9)).
The equation ax-j-l^ = c includes a number of subsidiary equations.
For instance ax^cx; thence a-^x^c-^x, and thence ax = cx. Similarly
i^^txc, and ^ = c^. The solution of the given equation will satisfy identi-
cally all these subsidiary equations.
(5) Particular Cases, There are two important particular cases of this
equation, when c = t, and when c = 0.
Firstly, c = i. Then cw? + te = i.
* Pointed ont to me by Mr W. E. Johnson.
30] ELIMINATION. 49
Hence a-^b^i.
But the only possible case of this subsumption is
a H- 6 = i.
Also ab 4 i, which is necessarily true.
Therefore finally, a + 6 = i, is the sole deduction independent of x.
Secondly, c = 0. Then cue + 65c = 0.
Hence a-^b^O, which is necessarily true.
Also ab 4 0. But the only possible case of this subsumption is aft = 0.
Therefore finally, a6 = 0, is the only deduction independent of w. If
the equation be written f(x) = 0, the result of the elimination becomes
/(»)/(0)=o.
These particular cases include each other. For if
cuV'{-bx = i,
then ~ {ax + b^) = 0,
that is ax + 6^ = 0.
And a + 6 = i is equivalent to ofc = 0.
(6) General Equoition. The general form ^{x) —i^{x), where ^ (a?) and
'^ (x) are defined in the same way as f(x) in § 29 (3), can be reduced to
these cases. For this equation is equivalent to
<l>(x)^(x) + ^(x)^fr{x) = 0.
This is easily proved by noticing that the derived equation implies
that is ^ (^) 4 ^ (^) » '^ (^) 4 ^ (^)>
that is <^ (a?) = ^ (x).
Hence the equation ^ (^) = '^ (^) can be written by § 29 (4) in the form
{* (*)^ « + * » ^ (i)} ^ + {<^ (0) ^(0) + ^ (0) ^ (0)} S = 0.
Hence the result of eliminating x from the general equation is
[<!> (t) ?(») + * (t) t (»)} {"^ (0) t (0) + * (0) t (0)1 = 0-
This equation includes the four equations
The reduction of the general equation to the form with the right-hand
side null is however often very cumbrous. It is best to take as the standard
form
ax + bx = cX'{-daD (1).
This form reduces to the form, ax + bx — c^ when c = d. For
cx + c^ = c(x-^x) = c.
w. 4
50 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
The equation is equivalent to the two simultaneous equations
cue = ex, 6S = d£ ;
as may be seen by multiplying the given equation respectively by x and x.
Let the equation ax = ex he called the positive constituent equation,
and the equation bx = c^ be called the negative constituent equation of
equation (1).
Taking the supplements
a + x^c-i-x, b-^x = d-^x.
Hence multiplying by x and x respectively
dx = cx, and bx=cbb.
So equation (1) can also be written
ax-^bx = cx-^dxi
and the two supplementary forms give
aa? + fe = ca? H- dx,
ax-^bx = cx-^ dx.
The elimination of x can also be conducted thus.
Put each side of equation (1) equal to z.
Then (ix-j-bx = z,
cx-^-dx^z.
Hence the following subsumptions hold,
a + b^z^db;
c-^-d^z^cd.
Therefore a-hb^cd,
c + d^ah.
Also similarly from the form, ax-hbx=^cx-\-dx,we find the subsumptions
d-^b^cd,
o-^d^ab.
These four subsumptions contain (cf § 31 (9), below) the complete result
of eliminating x from the given equation (1). The two supplementary forms
give the same subsumptions, only in their supplementary forms, but in-
volving no fi'esh information.
From these four subsumptions it follows that,
abed = abed = abed = abed = 0.
These are obviously the four equations found by the other method, only
written in a different notation.
30] ELIMINATION. 61
From these equations the original subsumptions can be deduced. For
abed = 0 can be written
oft 4 "" (^^),
and therefore ab^c-\-d.
Similarly for the other subsumptions.
Also it can easily be seen that the four subsumptions can be replaced
by the more symmetrical subsumptions, which can be expressed thus,
(The sum of any two coeflScients, one from each constituent equation)
^ (The product of the other two).
(7) IHscrimiruints. All these conditions and (as it will be shown) the
solution of the equation can be expressed compendiously by means of certain
functions of the coefficients which will be called the Discriminants of the
equation.
The discriminant o{ax = cx is dc + ac. Let it be denoted by A.
The discriminant of b^ = dx is bd + bd. Let it be denoted by B,
Then A and B will respectively be called the positive and negative
discriminants of the equation
ax + bx = cX'\'C^ (1).
Now il = ■" (ac + ac) = ac + ac,
and B = - (bd + bd) = bd + bd.
Therefore, remembering that y + ^ = 0 involves y = 0, z = 0, it follows that
all the conditions between the coefficients a, b, c, d can be expressed in
the form
This equation will be called the resultant* of the equation
aw + bx=:cx + dx.
It can be put into the following forms,
A^B, B^A, AB=^A, BA^B, A+B = B, B+A=A.
It is shown below in § 31 (9) that the resultant includes every equation
between the coefficients and not containing x which can be deduced fix)m
equation (1).
The equation ax-i-bx^cx + dx when written with its right-hand side
null takes the form
(8) Again let there be n simultaneous equations aiX=CiX,a^==c^,.,.,
On^^Cn^, and let Ai, A^, ... An be the discriminants of the successive
equations respectively; then their product AiA^... An is called the resultant
* Of. Schrdder, Algebra der Logik, § 21.
4—2
52 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
discriminant of the n equations. It will be denoted by 11(^1^), or more
shortly A. Similarly let there be n simultaneous equations biX = diX
b^=d^,...bnCb='dnX; and let Bi, B^-.-Bn be the discriminants. Then
B1B2 . . . £n is the resultant discriminant of the n equations. It will be called
n (Br) or B.
The n equations diX + 61^ = CiX + diX,
On^ + bnX =CnX-\' dnX,
involve the 2n equations just mentioned and conversely.
The functions A and B are called the positive and negative resultant
discriminants of these equations.
Now A = ill H- ila + . . . H- Any
5 = -Bi + JSj + . . . H- B^,
Hence AB = 1 ArB,.
Now any equation a,a! = CfX may be joined with any equation b^ = d^
to form the equation OfX + 6^ = c^a? + d^. Hence all the relations between
the coefficients are included in all the equations of the type,
ArB, = 0.
But these equations are all expressed by the equation
25 = 0.
This equation may therefore conveniently be called the resultant of the n
equations.
This is the complete solution of the problem of the elimination of a single
letter which satisfies any number of equations.
The single equation
Ax-^Bx = 0,
is equivalent to the 71 given equations.
It must be carefully noticed that in this algebra the distinctions of
procedure, which exist in ordinary algebra according to the number of
equations given, do not exist. For here one equation can always be found
which is equivalent to a set of equations, and conversely a set of equations
can be found which are equivalent to one equation.
(9) More than one Unknown, The general equation involving two un-
knowns, X and y, is of the type
axy + bay + c% + d^ = exy •{-fa^ -f gxy + Aay .
This equation is equivalent to the separate constituent equations,
axy = ftry, bay —foffy, etc. Let a constituent equation involving x (as
distinct from ^) be called a constituent positive with respect to x^ and
30] ELIMINATION. 53
let a constituent equation involving x (as distinct ftom x) be called a
constituent negative with respect to x. Thus, axy = exy^ is positive with
respect both to x and y ; bay ^^fxy^ is positive Avith respect to x, negative
with respect to y^ and so on.
Let A, B, Cy 1) stand for the discriminants of these constituents. Thus
A^ae-^ae, B = hf-v If, C — cg + ^, D^^dh-^dh, Then the discriminant
A is called the discriminant positive with respect to x and y, B ia the
discriminant positive with respect to x and negative with respect to y, and
so on.
The equation can be written in the form
(ay '\-by)x + (cy + dy) x=^{ey +/y)a? 4 {gy + hy) x.
If we regard x as the only unknown, the positive discriminant is
(ay + Wi {ey -^fy) + (ay + by) (ey +/y),
that is Ay + By.
The negative discriminant is Oy + Dy.
The resultant is (Ay-^ By) ( Cy + Dy) = 0 ;
that is AGy-^BDy = 0.
This is the equation satisfied by y when x is eliminated. It will be
noticed that A and C are the discriminants of the given equation positive
with respect to y, and B and D are the discriminants negative with respect
toy.
Similarly the equation satisfied by x when y is eliminated is
ABx-^ CDx = 0.
The resultant of either of these two equations is
ABCD = 0.
This is therefore the resultant of the original equation.
The original equation when written with its right-hand side null takes
the form
Axy'^Bxy-\-Cxy + Ds^ = 0 (1).
Again suppose there are n simultaneous equations of the above type
the coefEcients of which are distinguished by suffixes 1,2, ... n.
Then it may be shown just as in the case of a single unknown x, that all
equations of the type, ApBqCrDg = 0, hold.
Hence if A stand for AiA^ ... A^, and B for BiB^ ... B^, C for (7i(7, ... Cm
and D for AA ••• A, the resultant of the equations is ABCD = 0.
The n equations can be replaced by a single equation of the same form as
(1) above.
Also the equation satisfied by x, after eliminating y only is
ABx-^CDx=^0,
64 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IL
where A and B are the positive discriminants with respect to x, and C
and D are the negative discriminants. The equation satisfied by ^ is
ACy + BDy = 0,
where a similar remark holds.
(10) This formula can be extended by induction to equations involving
any number of unknowns. For the sake of conciseness of statement we will
only give the extension from two unknowns to three unknowns, though the
reasoning is perfectly general.
The general equation for three unknowns can be written in the form
{az H- a'z) xy + {bz 4- Vz) aiy + (cz + dz) xy + {dz + d!z) xy
= (ez + ez) xy + (fz -\-fz) xy-^(gz + g'z) xy-^{hz + Kz) s^.
Then, if A =a€ + ae, A' = ae + a'e, and so on, A is the discriminant
positive with respect to x, y, and z, and A' is the discriminant positive with
respect to x and y, but negative with respect to z] and so on.
If X and y be regarded as the only unknowns, then the two discriminants
positive with respect to x are
(az + az) {ez + e'z) + (az + a'z) (ez + e'z\
and (bz + b'z) (fz -{-fz) -h (Iz + b'z) (fz ^fz\
that is, Az->r A% and Bz + Rz.
Similarly the two discriminants negative with respect to x are
Cz + (7z, and Dz + Uz.
Hence the equation for x after eliminating y is
(Az-{-A'z)(Bz + Wz)x'\'(Cz-{-G'z)(I)z-\'D'z)x = Q,
that is (ABz -h A'Ez) x + ( CDz + CD'z) x = 0.
The result of eliminating z from this equation is
AA'BB'x + CCDD'x = 0.
Hence the equation for x after eliminating the other unknowns is of the
form, Pa? + Qx = 0, where P is the product of the supplements of the discri-
minants positive with respect to a?, and Q is the product of the supplements
of the discriminants negative with respect to. x.
The resultant of the whole equation is
AA'MGCDD' = 0,
that is the product of the supplements of the discriminants is zero.
The given equation when written with its right-hand side null takes the
form
AxyZ'\-A'xyz-\-Ba^Z'\'Fayz-\- Cxyz + Cxyz + l>c^z + Us^=^0.
The same formulae hold for any number of equations with any number
of variables, if resultant discriminants are substituted for the discriminants
of a single equation.
31] ELIMINATION. 60
(11) It is often convenient to notice that if
^ (a?, y, z, ...) ^-^{x, y, z, ...),
be an equation involving any number of variables, then any discriminant is
of the form
*lo,o,o,..J^Uo,o,.J + *Uo,o,..J^U,o,o,..J'
where % is substituted for each of the unknowns with respect to which the
discriminant is positive and 0 is substituted for each of the unknowns with
respect to which the discriminant is negative.
(12) The formula for the elimination of some of the unknowns, say,
UjV,w,..., from an equation involving any number of unknowns, a?, y, xr, ...
u,v,w,.,,, can easily be given. For example, consider only four unknowns,
^> y, ^> t, and let it be desired to eliminate z and t from this equation, so that
a resultant involving only x and y is left. Let any discriminant of the
• • ■ • .
equation be written D Q' l" *' * J , where either i or 0 is to be written ac-
cording to the rule of subsection (11). The equation can be written
{D(i t, I, %)xyz-hn(i i 0, i)xyz + I){i 0, i, i)xyz'^n(0, i, i, %)xyz
+ D{i 0, 0, %)xyZ'\-D (0, i, 0, %) xyz + D (0, 0, t, i) xyz + D (0, 0, 0, t) xyz] t
+ [D (%, %, i 0)xyz + 5(i, t, 0, 0)xyz-j-...]~t = 0.
Hence eliminating t, the resultant is
B(iiii)D{i^,iO)xyz + n(i,iO,{)n{iiO,0)xyz
+ 5(i,0,i,t)5(t,0,i,0)a:y^+...+5(0,0,0,i)5(0,0,0,0)Sp = 0.
Again eliminating z by the same method, the resultant is
5(i, i i %)D(i i i 0)5(t, i, 0, i)S(t, i, 0, 0)xy
4-5(i,0,t,i)5(i,0,i,0)5(i,0,0,t)B(i,0,0,0)a^
+ D (0, t/i, i) D (0, i, i, 0) D (0, t, 0, t) D (0, i, 0, 0) xy
+ 5(0, 0,i,i)5(0,0,i, 0)5(0, 0,0,t)2>(0,0,0,0)^y = 0.
It is evident from the mode of deduction that the same type of formula
holds for any number of unknowns.
31. Solution of equations with one unknown. (1) The solutions of
equations will be found to be of the form of sums of definite regions together
vrith sums of undetermined portions of other definite regions ; for example
to be of the form aH- Vift-h VjC, where a, 6, c are defined regions and Vi, Vj are
entirely arbitrary, including % or 0.
Now it is to be remarked that u(b-{- o), where u is arbitrary, is as
86 1*HE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. 11.
general as vjb + VjC. For writing w = Vji + v, (6 f 6c), which is allowable since
u is entirely arbitrary, then
u {h •\- c) = [vjb -{-v^^h -\-hc)] {6 + c}
= Vih + Vj (6c H- 6c)
= Vj) + VjC.
Hence it will always be sufEcient to use the form t^ (6 + c), unless v^ and v^
are connected by some condition in which case vji> + v^ may be less general
than w (6 H- c).
(2) ax = ca?.
Then by § 30, (7) Ax = 0.
Hence by § 26, Prop. VIII. x = Ax.
But instead of x on the right-hand side of this last equation, (x + vA)
may be substituted, where v is subject to no restriction. But the only
restriction to which x is subjected by this equation is that it must be
incident in A. Hence x-\-vA \& perfectly arbitrary.
Thus finally
x=-vA\
jre V is arbitrary.
(3)
bx^d^.
From subsection (2) ;
x = uB.
Hence
x=u-^B,
(4) Gw? + 6ac = ca; + cte ; where AB = 0.
From the equation (ix==cx,ii follows that x = uA ;
and from 6^ = d«, that x = v-^ B,
Hence uA = » + A
Therefore vA = 0.
Hence v = wA.
Finally, x = B + wA ;
where w is arbitrary.
This solution can be put into a more symmetrical form, remembering that
B-\'A=A.
For x = B{w-^w)'{-wA^Bw + w(A ■^B)^wA +wB.
Hence the solution can bo written
x = B-^wAA
x = A -j-wBj
31] SOLUTION OF EQUATIONS WITH ONE UNKNOWN. 57
Or a; = wA + wBy^
x = wA+wB.)
The first form of solution has the advantage of showing at a glance the
terms definitely given and those only given with an undetermined factor.
(5) To sum up the preceding results in another form: the condition
that the equations ax^cx, bx^da may be treated as simultaneous is
15 = 0.
The solution which satisfies both equations is
x = B + uA.
The solution which satisfies the first and not necessarily the second is
x^uA.
The solution which satisfies the second and not necessarily the first is
x = uB, that is x==u-{-B.
In all these cases u is quite undetermined and subject to no limitation.
(6) The case cw? + te = c, is deduced from the preceding by putting d = c.
Then A — ac-\-ac, B = bc-\-bc.
The solutions retain the same form as in. the general case.
The relations between a, b, c are all included in the two subsumptions
a-\-b^c^ab.
The case oo? + 6S = 0 is found by putting c == d = 0.
The equation can be written
(M? H- te = Oa? + 0^.
The positive discriminant is aO + aO, that is a, the negative is 6. The
resultant is oi = 0. The solution is
a? = 6 -I- lid,
(7) The solution for n simultaneous equations can be found with equal
ease.
Let X satisfy the n equations
OiX + biX = CiX + d^,
Oja? + ftjS = Cgfl? + d^,
ttnX+bnX = Cf^-i-dnX,
Then x satisfies the two groups of n equations each, namely
diX^CiXy a^=^c^ ... Of^^CfiiX]
and biX = diX, b^ = d^ . . . 6„a = d^ac.
= wA + wB,\
= wA + wB J
58 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. H.
From the first group
Hence x = uA ; where u is not conditioned.
Similarly from the second group
X = ViBi = v A = . . . = vJBn .
Hence x = vB, a; = » + jB ; where v is not conditioned.
Therefore uA = v + B.
Hence vA = 0, that is v = wA.
So finally the solution of the n equations is
x = B-{- wA =
x = A-{- wB
The group Oja? = CiX, eye = c^, etc. can always be treated as simultaneous,
and so can the group of typical form 6^ = d^^.
The condition that the two groups can be treated as simultaneous is
25 = 0.
(8) It has been proved that the solution B + uA satisfies the equation,
AxA- Bx=sO, without imposing any restriction on u. It has now to be
proved that any solution of the equation can be represented hy B-\-uA, when
u has some definite value assigned to it.
For if some solution cannot be written in this form, it must be capable
of being expressed in the form mB + wA •{-nAB.
But Ax^O, and AB = 0, hence, by substituting for x its assumed form,
nAB = 0. Thus the last term can be omitted.
Again, Bx=0; and AB = 0, hence B{m-{-B)(w + A) = 0; that is
mwB = 0. Hence m=p(w-\-B)y and therefore m=p + wB.
Therefore the solution becomes
x = mB-^wA = (p-j-wB)B-\-w{A-^B),
-pB-^-B-^-wA^B-^wA,
Thus the original form contains all the solutions.
(9) To prove that the resultant AB = 0, includes all the equations to be
found by eliminating x from
Ax + Bx = 0.
For a? = £ H- wA satisfies the equation on the assumption that AB = 0,
and without any other condition.
Hence AB is the complete resultant.
It easily follows that for more than one unknown the resultants found
in § 30 are the complete resultants.
32] SOLUTION OF EQUATIONS WITH ONE UNKNOWN. 69
(10) Subsumptions of the general type
aa + bx^cx + d^
can be treated as particular cases of equations.
For the subsumption is equivalent to the equation
cfl? + d» = (ca? + dS) (ax + 6^)
= dcx + bdx.
Hence A^ac + c" (ac) = ac + c==a + c,
-B=6d + d-(6d) = 6d + d = 6 + d,
A ^c{a + c) = ac,
B = d(b-\-d)^bd.
Therefore the resultant -4-8 = 0 is equivalent to fl^crf = 0. This is the
only relation between the coefficients to be found by eliminating x.
The given subsumption is equivalent to the two subsumptions
ax ^ ex J bx^dx;
that is, to the two equations
ex = acx, dac = bdx.
The solution of aa^cx is x = uA = w(a + c).
The solution of bx^dx \b x = U'\-B = u-\-bd,
The solution of ax + bx^cX'\-dx
is a? = -B + t^il =• Si + w (a H- c)
= uB + uA = M (a + c) + ubd.
The case of n subsumptions of the general type with any number of
unknowns can be treated in exactly the same way as a special type of
equation.
32. On Limiting and Unlimiting Equations. (1) An equation
^(^> y» ^> ••• 0 — V^C^* y> ^> ••• 0 involving the n unknowns x, y, z, ... t is
called unlimiting with respect to any of its unknowns (x say), if any
arbitrarily assigned value of x can be substituted in it and the equation can
be satisfied by solving for the remaining unknowns j/y z, ... t; otherwise the
equation is called limiting with respect to x. The equation is. unlimiting
with respect to a set of its variables x, y, z, ..., i{ the above property is
true for each one of the unknowns of the set. The equation is unlimiting
with respect to all its unknowns, if the above is true for each one of its
unknowns. Such an equation is called an unlimiting equation.
The equation is unlimiting with respect to a set of its unknowns
simultaneously f if arbitrary values of each of the set of unknowns can be
simultaneously substituted in the equation.
60 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IL
It is obvious that an equation cannot be unlimiting with respect to all its
unknowns simultaneously, unless it be an identity.
(2) The condition that any equation is unlimiting with respect to an
unknown x is found from § 30 (10). For let P be the product of the supple-
ments of the discriminants positive with respect to x and Q be the product
of the supplements of the discriminants negative with respect to x. Then
the equation limiting the arbitrary choice of x is, Px -f (2^ = 0. Hence if the
given equation be unlimiting with respect to x, the equation just found must
be an identity. Hence P= 0, Q = 0.
(3) The condition that the equation be unlimiting with respect to a set
of its unknowns is that the corresponding condition hold for each variable.
(4) The condition that the equation is unlimiting with respect to a set
x,y,z, ... of its unknowns simultaneously is that the equation found after
eliminating the remaining unknowns t^u^v, ... should be an identity. The
conditions are found by reference to § 30 (12) to be that each product of
supplements of all the discriminants of the same denomination (positive or
negative) with respect to each unknown of the set, but not necessarily of the
same denomination for different unknowns of the set, vanishes.
(6) Every equation can be transformed into an unlimiting equation.
For let the equation involve the unknowns x,y,Zj ... t: and let the resultant
of the elimination of all the unknowns except x be, Px + Qaj = 0.
Then a? = Q + uP, and if u be assigned any value without restriction, then
X will assume a suitable value which may be substituted in the equation
previous to solving for the other unknowns. Thus if all the equations of the
type Px+Qx = 0, be solved, and the original equation be transformed
by substitution of, x=^Q-\-uP, y=8 + vR, etc., then the new equation
between t^, v, ... is unlimiting.
(6) The field of an unknown which appears in an equation is the
collection of values, any one of which can be assigned to the unknown
consistently vrith the solution of the equation. If the equation be un-
limiting with respect to an unknown, the field of that unknown is said to be
unlimited ; otherwise the field is said to be limited.
Let the unknown be x, and with the notation of subsection (5), let the
resultant after eliminating the other unknowns be Px + Qx = 0. Then
a? = Q + uP. Hence the field of x is the collection of values found by
substituting all possible values for u, including % and 0. Thus every member
of the field of x contains Q; and P contains every member of the field,
since PQ = 0. The field of x will be said to have the minimum extension Q
and the maximum extension P.
33. On the Fields of Expressions. (1) Definition. The 'field
of the expression if){x, y, z, ... t)* will be used to denote the collection of
33] ON THE FIELDS OF EXPRESSIONS. 61
values ^hich the expression ^ (x, j/y Zy ... t) can be made to assume by
different choices of the unknowns Xyj/.z, ... t If ^ (x, y, z, ... t) can be
made to assume any assigned value by a proper choice of x, y, z, ... t, then
the field o{ <f>{x, y, Zy...t) will be said to be unlimited. But if ^(a;, y,Zy...t)
cannot by any choice of a?, y, Zy ...t, be made to assume some values, then the
field of <l> (xy y, z, ... t) will be said to be limited.
(2) To prove that
axyz ... t + bxyz ... i + ...kxyz ... t,
is capable of assuming the value a + 6 + ... + A. This problem is the same
as proving that the equation
axyz ...t-\-bxyz ...t-\-... •\-kxyz ...? = a + 6 + ...+&,
is always possible.
The discriminants (cf. § 30 (11)) are
il=a + a6...&, 5=6 + d6...^, ...K^k-\-ah ,..Tc.
Hence
il =a(6+c + ...&), -B = 6(a + c + ... +A?), ...Z^=i(a + 6 + c + ..•)•
Hence the resultant AB ... if = 0 is satisfied identically.
It is obvious that each member of the field of the expression must be
incident in the region a + 6 + c+...+i: for a + 6 + c + ... + A is the value
assumed by the expression when i is substituted for each product xyz...t,
xyz ... ty ... xyz ...t. But this value certainly contains each member of the
field.
(3) To prove that any member of the field of
axyz ... t-\-bxyz ...i + ... +J(^cyz ...t
contains the region abc ... A?.
For let <f>(Xy y, z,... t), stand for the given expression. Then the region
containing any member of the field of 0 (Xy yyZy...t) by the previous subsection
is a + E + c+... + fc. Hence the region contained by any member of the
field of <l>(XyyyZ, ...t) is abc.k. Hence combining the results of the
previous and present subsections
aH-6 + c + ... + A^^(a?, y, z, ...t)^abc ...k.
The field of (f>(x,yyZ... t) will be said to be contained between the
maximum extension a + 6 + . . . + A;, and the minimum extension ah ... k. '
(4) The most general form of p, where
a + 6 + c + ...■\-k^p^ahc...ky
is p^ahc... A; + w(a + 6 + c + ... +fc).
In order to prove that the fields of
^{Xy y, Zy...t) and abc ... A; -I- w(a + 6 + C + ... A;),
62 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IL
are identical, it is necessary to prove that the equation
<l>{x, y, z, ... t) = abc ... A? + w(a+ 6 + c + ... +i),
is unlimiting as regards u.
The equation can be written
axyz ...t-\- hxyz ... it + ... •\-kxyz ... t=ahc ...fci6 + (a + 6 + c + ...i)w.
The discriminants positive with resjpect to u are (cf. § 30 (11))
a(a + 6 + c.i.+A;) + a. ahc ...k, that is, a + abc... k,
and b-^dbc...k, c + dbc ...k, ...k + dbc ...k.
Their supplements are ^
a(6 + c + ...+A;), b(a + c-\-... + k), c(a-f 6 + .,. +i), ... i(a + 6 + c + ...).
Hence the product of the supplements is identically zero.
Similarly the discriminants negative with respect to u are
abc...k + d(a + b + ...-¥k\ abc ...k + b{d i-b+ ... +i),
and so on. Their supplements are a(b'\-c + ... +k), and so on. The
product of the supplements is identically zero. Hence (cf. § 32 (2)) the
equation is unlimiting with respect to u.
Thus* the fields of ^ (a?, y,^,...^) and of a6c ...A; + w (a + 6 + c+... + A:) are
identical and therefore without imposing any restriction on u we may write
^(^> y> z,...t) = dbc ...k + u(a-\-b + c+ ...-{-k).
(5) The conditions that the field of ^ (x, y,z,...t) may be unlimited are
obviously a6c... A; = 0, a + 6 + c + ... -f A:=t,
The two conditions may also be written
ahc ...k = 0 = dbc . . . i\
(6) Consider the two expressions
axyz ... f + bxyz ...i+ ... -{-kxyz ...i,
and c^uvw ...p'\-biuvw ...p+ ... hiuvw ...p,
not necessarily involving the same number of unknowns. Call them
^ (^, y, ^ . . . 0 8^d "^(^j v,w...p). The conditions that the field o{ <l>{x,y,z... t)
may contain the field of -^(w, v,w...p), i.e. that all the values which -^ may
assume shall be among those which 0 may assume, are ahc... k 4 (hbiCi ... Aj,
and a + 6 + c...+i^ai + 6i + Ci+...+Ai.
The two conditions may also be written
abc ...k^OfibiCi ... ^,
dbc ... k^dibiCi ...hi.
* Cf. SchnSder, Algehra def Lo^k^ Lecture 10, § 19, where this theoren) is dedaoed b^ another
proof.
33] ON THE FIELDS OF EXPRESSIONS. 63
(7) The conditions that the fields of <l>(x,y,z ...t) and -^(m, v, u;...|))
may be identical are obviously
abc ... k = (iibiCi ... Ai,
cbbc ... k^OribiCi ... A].
(8) To find the field o{f{x, y,z»..t), when the unknowns are conditioned
by any number of equations of the general type
<l>r(^, y, ^ ... t)^^lrr{x, y, z... t).
Write p = /(a?, y,z ...t); and eliminate x,y,z ..,t from this equation and
the equations of condition. Let the discriminant of the typical equation
of condition positive with respect to all the variables be Ar, let the dis-
criminant positive with respect to all except t be Br, and so on, till all the
discriminants are expressed. Then the resultant discriminants (cf. § 30 (8)
and (9)) of these equations are A=^H (Ar), -8 = 11 (Br), etc
Also let f(x, y, z .,.t) be developed with respect to all its unknowns, so
that we may write
p=^axyz ...t + bxyz ... i-\- ... .
The discriminants of this equation are pa+p^, pb-hpb, etc. Hence the
resultant after eliminating x,y, z ...tis
'{(pa+pa)A}-{(pb-\-pb)B}...=^0,
that is, {|)(d + Z) + p(a + Z)}{p(6 + 5)+j5(6 + 5)}...=0.
Hence |>(a + Z)(6 + £) ...+p(a + Z)(6 + £) ... = 0.
Thus (cf. § 32 (6)) the field of |> is comprised between
(a + Z)(6 + 5)... and aA-¥bB+....
But apart firom the conditioning equations the field of p is comprised be-
tween abc . . . and a + 6 -h c + . . . . Thus the effect of the equations in limiting
the field of j9 is exhibited.
The problem of this subsection is Boole's general problem of this algebra,
which is stated by him as follows (cf. Laws of Thought, Chapter ix. § 8) :
'Given any equation connecting the symbols x,y,...w, z,..., required to
determine the logical expression of any class expressed in any way by the
symbols a?, y... in terms of the remaining symbols, w, z, etc.' His mode of
solution is in essence followed here, w, z, ... being replaced by the coefficients
and discriminants. Boole however did not notice the distinction between
expressions with limited and unlimited fields, so that he does not point out
that the problem may also have a solution where no equation of condition is
given.
A particular case of this general problem is as follows :
Given n equations of the type ayX + bfX==CrX + d^,
to determine z, where z is given hy z^ex +fx.
Let the discriminants of the n equations be A and B, those of the
equation which defines z are ez + ez, fz -{-fz.
64 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
Hence the resultant is "" (eAz + eAz) ~ {/Bz -\-fBz) = 0,
that is (ef+fA '^eB)z'^ (?/"+/4 + cJS) 0 = 0; where AB = 0.
Hence z = (ef+fA ■\-eB)-\-u (eA -h/B).
Another mode of solution, useful later, of this particular case is as
follows :
The solution for x of the equations is a? = £ -f vA, x=^A + vB.
Substitute this value of ^ in the expression for z.
Then z^eB 4-/2 + veA + lfB^{eA +fA) v + (fB + eS) v.
It is easy to verify by the use of subsection (7) that this solution is
equivalent to the previous solution.
(9) An example of the general problem of subsection (8), which leads
to important results later (cf. § 36 (2) and (3)), is as follows.
Given the equation Aon/ + Bxy k- Gxy + Dxy = 0, to determine ay, ay,
Put z = ay, then by comparison with subsection (8) a = i, 6 = 0 = c = d.
Hence (a + A)(b + B)(c-\-C)(d-\-D) becomes BCD, and aA-\-bB + cC+ dD
becomes A.
Thus, remembering that ABCD^ 0,
xy = BOD-^-uA = A (BGD + u).
Similarly a!y = ACD + uB = B{ACD'^ u),
ay^ABD-\-uC^C{ABDJfu),
xy=-ABC + uD^D{ABd-hu).
Also xy + xy^BC +u{A+D) = (A-¥D){BC ■{-u],
ay + xy:r=A~D-\-u(B + C)^(B-\-C){AD + u}.
It is to be noticed that the arbitrary term u of one equation is not identical
with the arbitrary term u of any other equation. But relations between the
various w's must exist, since xy ■{- xy ■{■xy -{-xy = i.
(10) It is possible that the dependence of the value of an expression
f{x, y,z ...t) on the value of any one of the unknowns may be only apparent.
For instance if f{x) stand for x + x, then /(a?) is always % for all values of x.
It is required to find the condition that, when the values of y, ^, ... ^ are
given, the value of f(x, y, z ...t)iB also given.
For letf(x, y, z .., 0 = ^i + ^f%* where /i and /, are functions of y, z ... t
only. Then on the right-hand side either i or 0 may by hypothesis be put
for X without altering the value of the function.
Hence /i =/(a?, y...t)=fi.
Thus /i =/j is the requisite condition.
34] ON THE FIELDS OF EXPRESSIONS. 65
Let /(a?, y, z ...t)he written in the form
w{ayz ...t + byz.., i + ...) + x(a'yz ...t -\-Vyz ..A-^ ...),
then the required condition is a = a', 6 = V, etc.
34 Solution of Equations with more than one unknown.
(1) Any equation involving n unknowns, x, y, z ,..r,s,t can always be
transformed into an equation simultaneously unlimiting with respect to a
set of any number of its unknowns, say with respect to x,y, z For let
Pi be the product of the supplements of the discriminants positive with
respect to a?, and Qi the product of the supplements of those negative with
respect to x. Then (cf § 30 (11)) the resultant after the elimination of all
unknowns except x is,
Pl30 + QiX = 0.
•Hence we may write, x ^ Q^^ P^Xi^ Q^-\- P^Xi, where x^ is perfectly
arbitrary. Substitute this value of x in the given equation, then the
transformed equation is unlimiting with respect to its new unknown Xi.
Again, in the original equation treat x a& known, and eliminate all the
other unknowns except y.
Then the resultant is an equation of the form
{Rx + 8x)y + {Tx +Ux)y=- 0,
where ii, S, T, U can easily be expressed in terms of the products of the
supplements of discriminants of the original equation. The discriminants
in each product are to be selected according to the following scheme (cf.
§30(12)):
J2, s, r, u
+, -, +, -
+, +, -, -
X
y
Now substitute for x in terms of a?i, and the resultant becomes
P^y + Q^y = 0,
where P, and Q, are functions of x^.
Solving, y = Qs + P>ys = Qiy^ + P^y^ ;
where y, is an arbitrary unknown.
If this value for y be substituted in the transformed equation, then an
equation between a^i, y,, j? ... r, », t is found which is unlimiting with respect
to Xy and y, simultaneously.
Similarly in the original equation treat x,yaa known, and eliminate all
the remaining unknowns except z : a resultant equation is found of the type
{Vjxy^ V^y + V^xy -\-V^y) z + (Wixy -\- W^ + W^y ■\- W^y)z = 0;
W. 5
66 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
where the F's and W'& are products of the supplements of discriminants
selected according to an extension of the above scheme. Now substitute
for X and y in terms of Xi and y,, and there results an equation of the
where P, and Q, contain Xi,y^,
Solving, g^Q^ + P,^, = Q,e, + P, j,,
where 5, is an arbitrary unknown. Then substituting for z, the transformed
equation involving Xi, y^, ^, ... r^Syt is unlimiting with regard to x^, y^, z^
simultaneously.
Thus by successive substitutions, proceeding according to this rule, any
set of the unknowns can be replaced by a corresponding set with respect
to which the transformed equation is simultaneously unlimiting.
(2) If this process has been carried on so as to include the n — 1 un-
knowns x,y,z,..8, then the remaining unknown t is conditioned by the
equation Pn^ + Qn^ = 0; where Pn and Q^ involve a?!, y, ...««-! which are
unlimited simultaneously.
Solving for f, ^ = Qn + PJn = Qnin + ?n^ ;
where t^ is an arbitrary unknown.
Thus the general equation ia solved by the following system of values,
where a?i, y, ... *„ are arbitrary unknowns.
(3) The generality of the solution, namely the &ct that the field of the
solution for any variable is identical with the field of that variable as
implicitly defined by the original equation, is proved by noting that each
step of the process of solution is either a process of forming the resultant
of an equation or of solving an equation for one unknown. But since the
resultant thus formed is known to be the complete resultant (cf § 31 (9)),
and the solution of the equation for one unknown is known to be the
complete solution (cf. § 31 (8)), it follows that the solutions found are the
general solutions.
It follows from this method of solution that the general solution of the
general equation involving n unknowns requires n arbitrary unknowns.
(4) Consider, as an example*, the general equation involving two un-
knowns,
axy + bay + (% + ds^ = exy -¥ fxy ■{• gxy + h^.
Let A, B,C,Dhe its discriminants.
Then x = CD + -{lB)x, = ~(AB)x^-\- CDx,,
* Cf. Schroder, Algebra der T^flik, $ 22.
35] SOLUTION OF EQUATIONS WITH MORE THAN ONE UNKNOWN. 67
Also {Ax'\'Cx)y-\- (Bx + Dx) y = 0.
Hence
y=Bx + lJx-\- (Ax + Cx) y^^(Ax'^Cx)y^'\-(Bx + 3x)y^
= {(A+ABC)x,-^(G-i-ACD)x,]y,+ l(AB'\-AB~D)x,+(GD + WD^^
As a verification it may be noticed that the field of y as thus expressed
is contained between A-\-C and BD. This is easily seen to be true, re-
membering that ABCD = 0,
(5) The equation' involving two unknowns maybe more symmetrically
solved by substituting (cf. § 32 (5))
x = CD + -{AB)u = CDu'^'(AB)u,
y-=BD+-(AC)v=-BDv'\--(AC)v.
Then u and v are connected by the equation*,
A BGuv -f ABDuv + A CDuv + BCDuv = 0.
This is an unlimiting equation: thus either t^ or v may be assumed
arbitrarily and the other found by solving the equation.
Thus V = ABDu + BCDu + -(ABCu + ACDu)p,
or u = AGDv + BCDv + -(ABCv + ABDv) q ;
where p and q are arbitrary.
36. Symmetrical Solution of Equations with two unknowns.
(1) Schroderf has given a general symmetrical solution of the general
equation involving two unknowns in a form involving three arbitrary un-
knowns.
The following method of solution includes his results but in a more
general form.
(2) Consider any unlimiting equation involving two unknowns. Let
Ay By 0, D be its four discriminants. Then the equation can be written in
the form
Axy -\- Bxy^ Cxy -\- Dxy-=0 .....(a).
Now put X = a^uv + bjuv + Ciuv + diuv (13),
y^a^uv + biUv + c^uv + diXlv (7).
Since the equation (a) is unlimiting (cf. § 32 (2)),
* This equation was pointed oat to me by Mr W. £. Johnson and formed the starting-point
for my inyestigations into limiting and unlimiting equations and into expressions with
limited and unlimited fields. As far as I am aware these ideas have not previously been
developed, nor have the general symmetrical solutions for equations involving three or more
unknowns been previously given, of. §§ 35 — 37.
t Alffihra der Logik, Lecture xn. § 24.
5—2
68 THE ALQEBBA OF SYMBOLIC LOGIC. [CHAP. II.
Also since the fields both of x and y are unlimited, then (cf. § 33 (5))
aJbiCidi = 0 = OibjCidi = CLJb^c^ = a^biC^di.
Substitute for x and y from (fi) and (7) in (a), and write ^ (p, q) for the
«
expression
Apq + Bpq + Cpq + Dp^.
Then the equation between u and v is found to be
^(oi, (ii)uv'^(l>{bi, b^)uv-i-<l>(ci, Ci)uv + <f>(di, di)uv = 0 (S).
Equation (S) is the result of a general transformation from unknowns x
and y to unknowns u and v,
(3) If the forms (J3) and (7) satisfy equation (a) identically for any two
simultaneous values of u and v, then
Thus if the pairs (a,, Oj), (61, 6j), (Ci, Cj), (di, dj) be any pairs of simul-
taneous particular solutions of the original equation, then (/3) and (7) are
also solutions.
(4s) Assuming that (oj, a,) ... (di, d,) are pairs of simultaneous particular
solutions of (a), it remains to discover the condition that the expressions (/3)
and (7) for x and y give the general form of the solution.
This condition is discovered by noting that the solution is general, if
when X has any arbitrarily assigned value, the field of y as defined by
equation (a) is the same as the field of y as defined by (7) when u and v are
conditioned by equation (fi).
Now equation (a) can be written
(Ja?+ Cx)y-\-(Bx+^x)y-0.
Hence the field of y as defined by (a) is contained between the maximum
extension (cf. § 32 (6)) Ax + Cx and the minimum extension Bx + Dx.
Now let Axy Bx, Gxy Dx be the discriminants of (/8) considered as an
equation between u and v. Then
ila5 = aia; + aiS, Bx = b^x-\-ZxX, Oaj = <a« + c,^, Dx^^diX + diX.
But by § 33 (8) the field of y as defined by (7), where u and v are
conditioned by (fi) is contained between the maximum extension
diAx + b^Bg + c^Gx + d^Dx,
and the minimum extension
(a, + Ax)(b,'\-Bx){c, + Cx)(d,^-I>x):
that is, between the maximum extension
(oiOj + bibi + 0108 + didj) x + (OiOj + bib^ + Cip, + did,) x,
and the minimum extension
{oi + a,) (61 + 6s) (c, + Cj) (di + dj) a? + (oj + a,) (61 + 6,) (Ci -^ c») (^i + d,) ag.
35] SYMMETRICAL SOLUTION OF EQUATIONS WITH TWO UNKNOWNS. 69
If the field of y be the same according to both definitions, then
ctiaj + 6i6a + (hPt + diCk^ A (e),
aia, + 5A + CiC + di(/, = 0 (?)»
(ai + a,)(6i'f6i)(ci + c,)(^ + cZ,) = -B (i;),
(ai + a,)(ti + 6a)(C| + c)(d, + cZ,) = B (0).
These equations can be rewritten in the form
ctiO, + ti6, + CiCa + did^ = il ( €i ) ,
OiO, + 6i6j + CiCa + didj = 0 ((:,),
Mj + 6i6a + CjCj + dida = J8 (17,),
OiOj + ftiSj + CiCj + didj^Z) (^1).
It follows from their symmetry that if y be given, the field of a; as
defined by (0) and conditioned by (7) is the same as the field of ^ as
defined by (a).
By adding €1 and rju di+hi + Oi-^di — A +B.
Hence Oiii^dli = AB,
By adding (CO and (0,),
ai + 6i + Ci + di = (7 + Z).
Hence OibjCidi = CD,
Similarly, d,6jCa^ = -Z(7, ajb^o^^^BD.
Thus if the conditions between A, B,G, D o{ subsection (2) are fulfilled,
then the conditions between Oj, b^, C], di and a,, ft,, c,, c2, of subsection (2)
are also fulfilled.
Hence finally if (Oi, Oj), (61, 6,), (ci, c,), (dj, d,) be any pairs of simul-
taneous solutions of (a) which satisfy equations (€]), (fi), (971), (0i), then the
expressions (fi) and (7) for x and y form the general solution of equation (a).
(5) Now take one pair of coefficients, say Oi and a,, to be any pair of
particular simultaneous solutions of the equations
Axy + Bxy -^ Cxy + Dxy =^0 (/c),
and xy^A (X).
These two equations can be treated as simultaneous ; for the discriminants
of (X) are A^ A, A, A. Hence the complete resultant of the two equations
is _____
4(J8 + il)(a+il)(2) + il) = 0,
that is ABCD^Q]
and this equation is satisfied by hypothesis. Thus {k) and (X) can be
combined into the single equation
70 THE ATX3EBBA OF SYMBOLIC LOGIC. [CHAP. II.
that is, since AB = AG = 0,
ixy + Aary + Axy ■\-{D-\- A)xy = Q.
Any solution of this equation gives xy = A^ osy^B, xy^C, ay:^D', and
hence any solution is consistent with equations (cj), (fi), (rfi\ (tfj).
This equation is a limiting equation. By § 34 (5) it can be transformed
into an unlimiting equation.
Put x = A-\-k, y^A'\-l.
Then the equation becomes
Akl + ADkl-=0.
Let another pair of the coefficients, say bi and 6,, be choseo to be any
particular solutions of the equations
Aay + Bay -f Cxy + Dxy = 0,
a/y^B,
These equations can be treated as simultaneous; and are equivalent to
the single equation _ _
Bxy-^Bxy + {C'\-B)xy + Bxy = 0.
Any solutions of this equation give xy ^ A, aiy = B, xy ^ C, xy ^ D,
To transform into an unlimiting equation, put a?=JB + m, jr = J8 + n.
Then the equation becomes
Bmn + BCmn = 0.
Let another pair of the coefficients, say Ci and c,, be chosen to be any
particular solutions of the equations
Axy + Bxy + Cxy + Dxy = 0,
xy = G.
These equations can be treated as simultaneous; and are equivalent to
the single equation
Gxy-\-(B-\- G)aJy-\-Cxy + Gxy = 0.
Any solutions of this equation give xy ^ A, ay ^ B, xy = G, xy ^ D.
To transform into an unlimiting equation, put x = G+p, y^G+q,
Then the equation becomes
BCpq + Gpq = 0.
Let the last pair of coefficients, namely di and d,, be chosen to be any
particular solutions of the equations
Axy + Bay + Gxy + Dxy = 0,
xy = D,
These equations can be treated as simultaneous; and are equivalent to
the single equation
(A + D)xy + Dxy + Dxy + Dxy = 6.
/
35] SYMMETRICAL SOLUTION OF EQUATIONS WITH TWO UNKNOWNS. 71
Any solutions of this equation give
ay^A, xy:^B, xy^G, xy^D.
To transform into an unlimiting equation, put x^^D-^Vy y = D-h8,
Then the equation becomes
ADr8'hDr8 = 0.
If the coefficients Oi, as..<(2,, have these values, then the equations (e),
(?)» (v)j (^) are necessarily satisfied.
Hence finally we have the result that the most general solution of the
unlimiting equation
Axy-\- Bay + Cxy + Dxy = 0,
can be written
x = {A-\-k)uv + (B + m)uv + Cpav + Drav,
y'={A-¥l)uv + Bnuv + (C-\-q)uv -\- Dsuv ;
where u, v are arbitrary unknowns, and k and I, m and n, p and q, r and 8,
are any particular pairs of simultaneous solutions of
Akl +15101 = 0^
5mn + 50mr? =0,
Cpq-hBCpq^O,
T)r8 + ADf8 = 0.)
Let these equations be called the auxiliary equations.
The auxiliary equations can also be written,
A<l>{kj)^0, 5^(m, n) = 0, G<l>(p,q)^0, 5<^(r, 5) = 0.
(6) As an example, we may determine k, I, m, n, p, q, r, 8 so that the
general solution has a kind of skew symmetry; namely so that x has the
same relation to il as ^ has to D,
Thus put A: = 0, i = Z; m = JB, n = (7; q = G,p = B; « = 0, r = 5. These
satisfy the auxiliary equations. Hence the general solution can be written,
remembering that BG = JB, GB = C,
X =s Auv + Buv + Guv, x=^Auv + Buv + Guv + uv,
y = uv + Buv + Guv + Duv, y = Buv + Guv -f Duv.
Again, put A; = t, 1 = 0] m=0, w = i; |) = 0, 5^ = t; r = i, « = 0. The
solution takes the skew symmetrical form
a? = tiv + Buv + Cttv, T = Buv + Cut; + uv ;
y = ilttt; + UV + Duv, y = ilw + uv + Diiw.
As another example, notice that the auxiliary equations are satisfied by
k ^ Wy I ^w, m ^ Wy n =sWy p = w, q = w, r = w, 8 = w.
72 THE ALQEfiRA Olf SYMBOLIC LOGIC. [CHAP. IL
Hence the general solution can be written
x — {A 4 w)uv + {B + w)uv + Cwuv + Dviniv,
y=^{A-\-w)uv-\- Bwuv + (0 + w) uv + Dwuv \
where u, v and w are unrestricted, and any special value can be given to w
without limiting the generality of the solution.
(7) The general symmetrical solution of the limiting equation can now
be given. Let Aay + Bivy + Gxy + Dxy = 0 be the given equation.
By§34(5),put x = GD-\-{A '^B)X, y = BD-\-(A + C)Y;
where X and Y are conditioned by
ABGXY+A~BDXY+ ACDXY+ BCDXY^ 0.
The general symmetrical solution for X and Y is therefore by (5) of this
section,
X ^ {A -^-B + 0 '\- k)uv -\- {A + B -\-I) + m)uv + ACDpav + BCDfuv,
F=(il + 5 + C+Z)wi; + ABDnuv + (A + C + D + q)uv + BCDsuv ;
where k,l; m,n; p, q; r, 8 are any simultaneous particular solutions of the
auxiliary equations
ABCkl + ABCDkl^O,\
ABDmn + ABGDmh = 0,
ACDpq^-ABGDpq^O,
BGDrs + ABGDfs = 0. j
(8) As a particular example, adapt the first solution of subsection (6)
of this section. Then a general solution of the equation is
a? = C5 -f (il + BG) uv -\-{B + AD)uv-\- AGD uw,
y = B5 + (4 + C) w + ABDm + {G + AD)uv + BGDuv.
(9) If a number of equations of the type,
V^i (^» y) = xi (^> y\ V^> (^» y) = x» (^> y)» etc.,
be given, then (assuming that they satisfy the condition for their possibility)
their solution can be found by substituting their resultant discriminants
(cf. § 30, (8), (9)) for the discriminants of the single equation which has
been considered in the previous subsections of this article,
(10) The symmetiical solution of an equation with two unknowns has
been obtained in terms of two arbitrary unknowns, and of one or more
unknowns to which any arbitrary particular values can be assigned without
loss of the generality of the solution. It was proved in § 34 (3) that no
solution with less than two unknowns could be general. It is of im-
portance in the following articles to obtain the general symmetrical
■(^)-
36] Johnson's method. 73
solution with more than two arbitrary unknowns. For instance take three
unknowns, u, v, w (though the reasoning will apply equally well to any
number). Let the given unlimiting equation be
Aon/ + Bay -^ Cxy -\- Dxy = 0 (a).
Put
a? = tti uvw + bi uvw + Ci uvw + di uvw ^
+ a^uvw + hlmw + c^uvw + diuvw,
y = a2UVW'\- +d^uvw
'\-(iiuvw-\- •\-d^umv.
Consider a? as a known, then the maximum extension of the field of y as
defined by (a) is -4 a? + Cfe, and its minimum extension is Sr + Dx,,
Also the maximum extension of the field of y as defined by {fi) is
SaiOs.ar+SaiOs.^c, and its minimum extension is n(«i + ai)a?4-n(ai + aa).;».
Hence, if {fi) is the general solution of (a), the following four conditions
must hold
SaiOasil, SoiOjssJS, XdjOi^^C, Soidi^D.
Also Oi, a,; 6i, 6a ••• d^\ d^\ must be pairs of simultaneous solutions of the
given equation (a).
36. Johnson's Method. (1) The following interesting method of
solving symmetrically equations, limiting or unlimiting, involving any
number of unknowns is due to Mr W. E. Johnson.
(2) Lemma, To divide a + 6 into two mutually exclusive parts x and y,
such that x^a and y^h.
The required conditions are
These can be written xy + ctary + hxy + (a + 6) aiy = 0.
Hence by § 34 (5), a? = a6 + aw = a (6 + w),) x, v
y=a6 + 6t; = 6(a + t;);j ^ ^'
where a6 (t^t; + uv) = 0 (2).
Solving (2) for v in terms of w, by § 31 (5), v = a&u + (a + 6 + u)t(;.
Substituting for v in (1) and simplifying,
a? = a (6 + w), y = 6 (a + u).
(3) Let the equation, limiting or unlimiting, be
Axy •}- Bay ■{- Cxy ■{- D^ ^ 0 (3).
The resultant of elimination can be written A + B + C + D^^i.
Also xy -^r^ and ay + xy are mutually exclusive, their sum
= t = 4 + J8 + (7 + A
and obviously fi'om the given equation xy-{-xy^A+D,a!y + xy:^B + C,
74 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IL
Hence by the lemma
xy + c^==(A-{D)(BG'^u\ xy + ccy = {B + C) (AD + u) (4).
The course of the proof has obviously secured that u does not have to
satisfy some further condition in order that equation (4) may express the
full knowledge concerning xy-\-xy and xy -f %, which can be extracted from
equation (3).
Also, as an alternative proof of this point, § 33 (9) secures that equation
(4) represents the complete solution for these expressions.
Again, by equations (4) xy + xy^^ BC + u, hence xy 4 BG + u.
Also by equation (3), xy 4 A. Hence by § 28, Prop. XIV., xy^A (BC + u).
Similarly ^^D (BC! + u).
Therefore by the lemma and equations (4) and simplifying
xy^A(BG^-u)(D-^p\\
c^ = D(BC-\'u)(A+p).\ ^^^•
Also, as before, it follows that equations (5) are a complete expression
of the information respecting xy and sty to he extracted from equation (3).
Similarly ay=-B (15 + u)(d+ g),
xy=^C(AD+u)(B + q)
Adding appropriate equations out of (5) and (6),
x^A(BO + u)(5+p)+B (AD + u)(C + q\)
y = A(BC^u)(D+p) + C(AD + u)(B + q).\
This symmetrical solution with three arbitraries is the symmetrical
solution first obtained by Schroder (cf. loc. cit.).
(4) A simplified form of this expression has also been given by Johnson.
For A (BC + u)(D + p)=^A (BCu + u) (D+p\
and B (AD -\-u)(^ + q) = B (ADu + u) (C' + g).
Hence
a; = u {AD + Ap + BAD(G + q)] + u {ABG(D +p) + JSO + Bq]
=^u{AD + Ap+BD(G + q)]'\'u{AD(D + p) + BG+Bq\
= A(d+u)(D + p) + B(D + u)(0'\'q).
Similarly y^A(B + u)(B+p) + C(D'hu)(B + q).
(5) This method of solution can be applied to equations involving any
number of unknowns. The proof is the. same as for two unknowns, and the
headings of the argument will now be stated for three unknowns.
Consider the equation
Axyz + Bxyz + Gxyz + Dxyz + A'xyz + B'afyz + G'xyz + Ds^z = 0. . .(1).
ary^B(AD + u)(G-^q\\
xy=^C(AD+u)(B + q)i ^^^'
37] SYMMETRICAL SOLUTION OF EQUATIONS WITH THREE UNKNOWNS. 75
The resultant isA+B-hC + D + A' + R + CT + D'^i.
Also from (1) xyz + ayz + xi/z + xyz ^A+D-\-B-{-Cy
xyz + (tyz + xyz + xyz :^B + C+ A' + D'.
Hence by the lemma, cf. subsection (2)
xyz-\-aiyz + xyz + 3^z=^(A'{'D+ B' + C') {BCA^U + 8)A
xyz + xyz + xyzi-£yz^(B+C + A'-{-iy)(ADB'G' + 8).) ^ ^'
Again, from (2) and (1),
xyz + xyz 4 (B' + C) {BCA'U + «),
xyz 4- iry-j 4 (il + D) (BCA'S^ + a).
Hence from the lemma, cf. subsection (2), and simplifying,
xyz + xyz=^{R + C) {BCAU + a) {AD + m)\
xyz+xyz^{A+D){BCA'D''\-8)(BV'+w).\ ^'^^^
Similarly
xyz ^-xyz^iB+G) {ADWG' + s) (A'& + n),)
xyz+xyz^(A' + iy){ADBV' + -8)(BC-^v)] ^ ^'
Again, fix)m equations (3) and (1),
xyz^A {BGA'D' + a) {B'C + w), a^z^D {BCA'U + a) (S'C + w).
Hence by similar reasoning to that above
xyz i=:A(BCA'5'+ a){B'Cr+ w) (B + *g), c^z = D (BGA'D'^ a)(B'C'^ in)(A + g).
Similarly,
xyz = B'(A'BCD'+ a)(AD + m)(G +p), xyz = Cr{A'BC5'+a)(AD + m)(5'+ p),
«y^ = C(AB'CD + 5)(Z'5' + w)(5 + 0, xyz^B (ABG'D + 5)( J '5' + n)(fi^t\
xyz = ^' (15'C'5 + a) (BG + ») (D' + 1), xyz = D' {ABV'D + 5)(5a + n) (J' + Z).
By adding the appropriate equations we determine Xy y, z.
This method is applicable to an equation involving n unknowns, and in
this case the solution will involve 2**— 1 arbitrariea
37. Symmetrical Solution op Equations with three unknowns.
(1) Consider the unlimiting equation
Axyz + Bayyz + Gxyz + Dl^z
+ A'xyz + Rxyz -h G'xyz + ffxyz =: 0 (a).
The conditions that the equation is unlimiting are (cf § 32 (3)),
ABGS=0 = 'AnffC'D'=^AA'CC'=:BRI)Ty = AA'B^^
Let the left-hand side of (a) be written <^ (x, y, z) for brevity.
> (fi)
76 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
Let the general solution of (a) be
X = diuvw + biuvw + Ciuvw + diuvw
+ Qiuvw + biUvw + Ci'uvw + di'uvw,
y = a^vw + h^uvw + c^uvw + d^uvw
4- a^uvw + b^'uvw + c^uvw + d^uvw ;
where m, v, w are arbitrary unknowns.
By substituting for x, y, z from equations (13) in equation (a) the con-
ditions that ()8) should be some solution of (a) without restricting u, v, w are
found to be,
<^(ai, ttj, a,) = 0 = <^(6i, 6„ 6,) = <^(Ci, c«, p,) = <^(di, cJ,, d,)
= 0 (oi', Oa', flsO = ^ (V, ^s', bi) = ^ (ci', c,', c,') = ^ (di', d,', d,0-
Thus the corresponding triplets of coefficients must be solutions of the
given equation.
(2) It remains to find the conditions that (fi) may represent the general
solution of (a). Eliminate z from (a), the resultant is
A A' xy + BSix^ + GC'xy + DU^ = 0.
By § 35 (10), the conditions that the first two equations of (fi) should be
the general solution of this equation are .
Similarly eliminating y from equation (a), the resultant is
ABxz + A'B'xz + CDxz + C'D^^ = 0.
The conditions that the first and third of equations {fi) should form the
general solution of this equation are
Lastly, eliminating x from equation (a), the resultant is
ACyz + A'C'yz + BSyz + RUyz = 0.
The conditions that the second and third of equations (fi) should form
the general solution of this equation are
ta^^A + G, -la^^A'^-G', Xd^=^B + D, la^^B'^-U.
(3) Again, if y and z be conceived as given, the field of ^ as defined by
equation (a) is contained between the maximum extension
Ayz + A'yz + Byz + Eyz,
and the minimum extension
Cyz + G'yz + Dyz + D'yz.
But (cf § 33 (8)) the field of d; as defined by equations (/9) is contained
37] SYMMETRICAL SOLUTION OP EQUATIONS WITH THBEE UNKNOWNS. 77
between the maximum extension Soi (o^ + a^y) {a^z 4 a^), and the minimum
extension 11 {a, + a^ + a^ + a^ + a^) ; that is, between the maximum ex-
tension
SoiO^ . yz + SaiOaOs . yz -j- ^aia^ . yz + ^a^^ . yz
and the minimum extension
n (oi + a, + a,) . y^ + n (oi +a, + a,) . y^
+ n (Oi + a, + a,) . y^ + n (oi + ttj + a,) . y0.
Hence since the extensions as defined by (a) and (y3) must be identical,
we find by comparison
The symmetry of these equations shows that, if z and x be conceived as
given, the field of y as defined by (a) is the same as that defined by {ff), and
that if X and y be conceived as given, the same is true for z.
By adding the appropriate pairs of this set of equations it can be seen
at once that these eight conditions include the twelve conditions of sub-
section (2).
Hence finally equations ()8) form the general solution of equation (a), if
the triplets o^, a,, a,; 6i, 6„ t,; ...; di\d^,d^\ are any simultaneous sets of
solutions of the given equation which satisfy the eight conditions above.
(4) Now following the method of § 35 (5), let Oi, a,, a, be any
particular simultaneous solutions of the equations
<^(^, y, ^) = 0,
and xyz = A.
These equations can be treated as simultaneous and are equivalent to
the single equation
Axyz + (jB 4- A)xyz + ((7 + A)xyz + (i) + A)xyz
+ (A' + A)xyz + (B'-hA)a^ + (C' + A) xyz + (U + A) xyz = 0.
This equation is in general a limiting equation. It can be transformed
into an unlimiting equation by writing (cf. § 32 (5))
x^{C + A)(C''^,A)(D'\-A)(D'+A) + ''(AA'BB')p,,
y = (B + A)(R+A){D + A)(D' + A) + -(AA'BB')p,,
z = (A' + A)(F -^ A)(C' + A)(D' -^^ A) ^-(ABGD)p,,
The conditions in subsection (1) that the original equation may be un-
limiting reduce these formulsB of transformation to
x = A-rpj, y^A+Pi, z = A+p^.
Then Pi,pi,pz satisfy the unlimiting equation
^PiP%Pt + A Bpip^pt + A GpiPiPt + A Dp^p^pi
+ AA'p^p^Pi + ARp^pip:, + A Cfp^p^p^ + A lypip^ps = 0 (1).
78 TUB ALGEBRA OF SYMBOLIC LOGIC [CHAP. IL
Similarly put 61 = 5 + ^i, 62 = ^ + Ja* &» = ^ + }», where q^, g„ 9, satisfy
the unlimiting equation
ABq^q^^ + 5?,?,?, + BCq^ q^q^ + BDq^q^qi
+ BA'q,q,q, + BRq.q^q, + BG'q.q^q, + Bj5'q,qjqs=- 0 (2).
Similarly put Ci = (7 + fi, Cj = (7 + rj, 03 = C+ r,, where ri, 7*a, r, satisfy
the unlimiting equation
A Cr^r^r^ + BCi\fir^ + Cvir^r^ + CDfifin
+ GA'r^r^f^ + Gffnr^r^ + CG%r^f^ + GD^f^f^f^ = 0 (3).
And so on for the remaining triplets of coefficients, putting
rfi == D + «i, d, = 2) + 5,. d, = i) + «s,
61' = -B' + 5/, h^ = E + 5/, 6,' = 5' + 5/,
c/= C' + r/, c' = C' + r/, c,' = G' + r,\
And the sets «i, «,, «,; pi'yp^* Pz\ ...; «/, «j', «s'; satisfy unlimiting equa-
tions formed according to the same law as (1),(2), (3). These other equations
will be numbered (4). (5), (6), (7), (8).
Let the equations (1)...(8) be called the auxiliary equations. When the
coefficients Oi, Os, a,; 61, 6„ &,; ...; dr!, d^, d^ have the values here assigned,
the eight equations of condition of subsection (3) are identically satisfied.
(5) Hence the general solution of the equation
Axyz-h Bayz+ Gxyz + Dxyz
+ A'scyz + B^aryz + G'xyz + Uxyz = 0,
is given by
ar = {A +pi) uvw + (J5 + q^uvw + Gr^uvw ^-Ds^vw
4- {A' + 1)/) uvw + (-B* 4- jiO wvw; + C'rxuvw + D'siuvw,
y-(A -^ P2) uvw + Bq^uvw + ((7 + ri)uvw + DsjUvw
+ (-4' +pi') t^vty + Bqiuvw + (C + r,') ttt;t(; + Ua^uvw,
z=^(A -hpt)uvw + (jB -h g,) uvw + ((7 + r,) ww-f (2) + «,)ui«i;
A'p^uvw + B'q^'nvw + G'r^uvw + D's^uvw ;
where pi,ps,p,; 9i> 92* 9s; •••; *i', «/, V; respectively are any sets of par-
ticular solutions of the auxiliary equations (1), (2)... (8) These equations
can be wiitten. _
^^(Pi,p»,p») = 0 (1),
^^(?i,9».?s) = 0 (2).
(7*(n.r2,r,) = 0 (3),
B<^(«i,«2, «,) = 0 (4),
37] SYMMETRICAL SOLUTION OF EQUATIONS WITH THREE UNKNOWNS. 79
Ay(p^,p,\pO==0 (5),
5^^ (?/,?/,?/) = 0 (6),
C>(r/, r;, r/) = 0 (Y),
B'^ (C *;, O = 0 .(8),
where <f> (x, y, z) stands for the left-hand side of the given equation.
It will be observed that we may put
where t, U, t^ form any particular solution of the given equation,
^ {^. y. ^) = 0.
(6) The general solution of a limiting equation involving three un-
knowns is found, first by transforming it into an unlimiting equation accord-
ing to § 32 (5), and then by applying the solution of subsection (5) of the
present section.
(7) The method of reasoning of the present section and the result are both
perfectly general. Thus the general equation involving three unknowns can
be solved with a redundaut unknown by the method of § 35 (10). Then by the
method of the present section the equation involving four unknowns can be
solved in a general symmetrical form. And the auxiliary equations will take
the same form : and so on for any number of unkuowns.
(8) As an example consider the equations
yz = a, 2x^ by xy = c.
These equations can be combined into a single equation with its right-
hand side zero by finding their resultant discriminants (cf. § 30 (9)).
The discriminants, cf. § 30 (1 1), of the first equation positive with respect
to X and y are a and a.
The discriminants of the first equation positive with respect to x and
negative with respect to y are a and a.
The discriminants negative with respect to x of the first equation are the
same as those positive with respect to x.
Hence the following scheme holds for the discriminants :
Constituent
xyz
xyz
xyz
xyz
xyz
xyz
xyz
xyz
Ist Equation ...
a
a
a
a
a
a
a
a
2nd Equation...
h
b
b
b
b
b
b
b
3rd Equation...
c
c
c
c
c
c
c
c
Resultant |
Discriminants/
abc
die
abc
abc
abc
abc
cUc
1
^hc
80 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. II.
The resultant is
(a + 6 + c)(a + 6+c)(a + 6 + c)(a + 6 + c)(a + 6 + c) = 0;
that is abc + abc + abc = 0.
This equation can be solved for c in terms of a and 6.
The positive discriminant is "" (ah + ab), that is ofe + ofc, the negative
one is a + ^.
The resultant is ab (ab + ab) = 0, which is identically true.
The solution is c = ab + u (ab + ab) = ab + v^.
Hence the solution for x is found from
a? = (a + 6 + c) (a + 6 + c) + w {abc 4- a6c 4 ohc + ate) = 6 + c + uoSic
= 6 + c + ua.
Similarly y = c + a + t;6, z = a-\-b'\-wc\
where u, v, w satisfy three unlimiting equations, found by substituting for
X, y, z in the original equations.
Thus substituting in yz = a, we obtain
a +-60 + abv + acw + bcvw = a ;
that is a-\-bc + bcvw = a, that is ofcc + abcvw = 0.
But abc = 0. Hence the equation becomes abcvw = 0.
Similarly dbciuu = 0 = dbcuv.
ft
Thus w, V, ti; satisfy the equation abc (ttv + vw-\- wu) = 0.
Comparing this with the typical form (a) given in § 37,
I=a6c = 5 = C = 2', D^O^R = C'^D\
Also a particular solution of the given equation is u = v = w=^0. , Hence
from subsection (5)
u=(a'\-b + c)(UVW+UVW+UVW)+UVW,
v=-{a + b + c){UVW+UVW+UVW)-{-UVW,
t£; = (a + 6 + c)(?7FF+FFTr+ UVW)+UVW.
Hence the general solutions for x, y, z are
a? = 6 + c + wa=6 + c + aUVW,
y = c + a + t;5 = c + a + VUVW,
where tT, F, Yf are arbitrary unknowns.
38. Subtraction and Division. (1) The Analytical (or reverse) pro-
cesses, which may be called subtraction and division, have now to be
discussed,
38] SUBTRACTION AND DIVISION. 81
Let the expressions a — b and a -^ 6 satisfy the following general conditions:
I. That they denote regions, as do all other expressions of the algebra ; so
that they can be replaced by single letters which have all the properties of
other letters of the algebra.
II. That they satisfy respectively the following equations :
(a — 6) + 6 = a ; (a -f 6) x 6 = a.
(2) Let X stand for a — 6. Then x is given by the equation
x + b=sa.
The positive discriminant is ai + al, that is a, the negative ia ab-\-ab.
The resultant is a (ab + a6) = 0, that is ab — O]
hence b^a.
The solution is a; = a6 + a6 + wa = a6+t4a.
Hence for the symbol a — b to satisfy the required conditions it is
necessary that b^a. Furthermore negative terms in combination with
positive terms do not obey the associative law. For by definition,
6 + (a - 6) = (a - 6) + 6 = a.
Also since b^a, and therefore 6 + a = a, it follows that
(6 + a) — 6 = a — 6 = a6+ tia.
Therefore 6 + (a — 6) is not equal to (6 + a) — 6.
This difficulty may be evaded for groups of terms by supposing that all
the positive terms are added together first and reduced to the mutually
exclusive form of § 27, Prop. X. Such groups of terms must evidently be
kept strictly within brackets. It is to be further noticed that the result of
subtraction is indeterminate.
(3) Again, for division, let a: = a -r 6.
Then bx = a.
The positive discriminant is ah -f-a6, the negative ia aO + aO, that is a.
The resultant is a {ab + a6) = 0 ; that is ab = 0.
Hence a^b.
The solution is x = a + v (ab + ab) = a -f- vdb = a + vfc.
Factors with the symbol of division prefixed are not associative with
those with the symbol of multiplication prefixed (or supposed).
For 6 (a -r 6) = (a -r- 6) 6 = a, by definition.
Also since ah^a,
(6a) -f-6 = a-r6 = a + t;6.
This difficulty can be evaded by suitable assumptions just as in the case
of subtraction. The result of division is indeterminate.
w. 6
82 THE ALGEBRA OF SYMBOLIC LOGIC. [CHAP. IT.
(4) Owing to these dif&culties with the associative law the processes of
subtraction and division are not of much importance in this algebra.
All results which might depend on them can be obtained otherwise*. They
are iiseful at times since thereby the introduction of a fresh symbol may
be avoided. Thus instead of introducing a?, defined by a? + 6 = a, we may
write (a — 6), never however omitting the brackets.
Similarly we may write (a -r b) instead of a?, defined by bx = a.
But great care must be taken even in the limited use of these sjrmbols
not to be led away by fallacious analogies.
For (a — 6) = at + wa ; with the condition b^a.
But {(a + c) - (6 + c)} -(a^c)-(b + c) + u(a + c)
= abc + u(a + c).
These two symbols are not identical unless
abc = ab, and a-^c^a.
From the first condition c^d + b,
that is ac^ai)
4 ^ 5 since ab^b.
From the second condition ac = c.
Hence from the two conditions c^b.
Again, (a -i- 6) = a + v6 ; with the condition a 4 ^•
But {(ac) -T- (be)} =ac + v -{be) = ac + 1; (6 + c).
These two symbols are therefore not identical unless
(ic = a, and b +c='ab.
From the second condition bc = a + b = b, hence b^c. This includes the
first condition which can be written a^c. But a 4 6. Hence the final
condition is b^c.
(5) It can be proved that
-(a-6) = (a^6), -(a-6) = (a~6).
For both (a — b) and (a -^ b) involve the same condition, namely 6 4 a, or
as it may be written a^b.
Again, a — 6 = a6 4-t;a.
Therefore ~ (a— 6) = (a + v) (a + 6) = a + vb.
But (a-i-b) = a + vh.
.Therefore the two forms are identical in meaning.
Similarly " (a -h 6) = (a - 6 ).
♦ First pointed out by Sehrikier, Der OperationkreU dea LogikkalkUlt, Leipsic, 1877.
CHAPTER III.
Existential Expressions.
39. Existential Expressions. (1) Results which are important in
view of the logical application of the algebra are obtained by modifying the
symbolism so as to express information as to whether the regions denoted by
certain of the terms either are known to be existent (i.e. the terms are then
not null), or are known not to include the whole of space (i.e. the terms are
then not equal to the universe). If this information is expressed the terms,
besides representing regions, give also the additional information, that they
are not 0, or are not i. When this additional existential information is being
given let the symbol = be used instead of the symbol = ; and let the use of =
be taken to mean that, in addition to the regions respectively represented by
the combinations of symbols on either side of it being the isame, the exist-
ential information on the right-hand side can be derived from that on the
left-hand sida
The B3rmboli8m' wanted is one which will adapt itself to the various
transformations through which expressions may be passed. If all regions
were denoted by single letters, it would be possible simply to write capital
letters for I'e^ons known to exist, thus X instead of x, and then the
information required, namely that X exists, would be preserved through all
transformations. Thus X at once tells us that X exists and that X does
not embrace all the universe %. But this notation of capitals is not sufficiently
flexible. For instance it is not possible to express by it that the region ai
exists : this requires that a exists, that b exists, and in addition that they
overlap, and this last piece of information is not conveyed by AB,
The merit of the symbolism now to be developed is that the new symbols
go through exactly the same transformations as the old symbols, and thus
two sorts of information, viz. the denotation of regions and the implication
of their existence, are thrown into various equivalent forms by the same
process of transformation.
(2) Any term x can be written in the form xi. Now when the fact has
to be expressed that x is not null, let i be modified into j*; so that xj expresses
that X exists, the j being added after the symbol on which it operates.
6-2
84 EXISTENTIAL EXPRESSIONS. [CHAP. III.
Furthermore any term x can be written in the form a? + 0. Now when
the fact has to be expressed that x does not exhaust the whole region of
discourse, that is to say is not i, let the 0 be modified into a>. Then
x + co expresses that x is not equivalent to i.
Let any combination of symbols involving j' or o) be called an existential
expression.
Thus j may be looked on as an affirmative symbol, giving assurance of
reality, and o) as a limitative symbol restraining from undue extension.
They have no meaning apart from the terms to which they are indissolubly
attached, the attachment being indicated by brackets when necessary, Le. by
(xj) and by (x + <o).
It is to be noted that aj or x + o) can be read off as assertions : thus xj
states that x is not 0, a; + a> that x is not ^.
(3) The sjrmbol xy .j will be taken to mean that j operates on xy^ so that
xy exists. Thus xy ,j implies xj and yj ; but the converse does not hold.
The mode of attachment of j to the term on which it operates has some
analogy to multiplication as it obtains in this algebra. Thus
^'yj'j = ^'j>
though xj.yj is not equivalent to xy.j s^ far as its existential information
is concerned.
Again, if a;, y, z, u, ... represent any number of regions, then
(xyzu..,)j = {ay.yj.zj...)j;
but the final j cannot be omitted, if the existential information is to be the
same on both sidea
(4) The distributive laws have now to be examined as regards the mul-
tiplication and addition of existential expressions.
Consider in the first place the expression {x + y)j.
Now if a? = 0 and y = 0, then a? + y = 0. Hence (x + y)j implies either
aj or yj or both. Thus we may adapt the symbolism so as to write
(x + y)j = xji + yji;
where the suffixes of the /s weaken the meaning to this extent, that one of
the /s with this suffix is to hold good as to its existential information but
not necessarily both. We define therefore
^i + yji + ^ji-^ ...
to mean that one of the terms at least is not 0.
The other formal properties (cf. subsection (3)) of j evidently hold good,
retaining always this weakened meaning.
The only point requiring notice is that xjij = xj ; for j, has the same
meaning as ^* in a weakened hypothetical form.
Further (xj + y)j = xjj^ + yj, ^xj + y;
for the ji can be omitted, since it is known that x exists.
39] EXISTENTIAL EXPRESSIONS. 85
In using the multiplication of existential symbols the dots (or brackets)
must be carefully attended to. For instance
(os + y). zj^x.zj + y . zj.
But Oc + y)z .j = {xz + yz)j = xz .j^ + xz .j^.
In the first expression {x + y).zjf the j simply asserts that z exists; in
the second expression it asserts that (x + y)z exists.
Again, xy + z=^{x + z)(y + z).
Also xy .j -¥ z implies {x-^- z){y-\'Z) ,j.
But though xyj +z = (x + z){y + z) .j ; the left-hand side gives more
definite information than the right-hand side. For
(x-{-z){y + z) .j^xy.ji +zj^.
Also xy ,j implies xj, yj.
Hence ay .j + z = (xj 'hz)(yj + z)j = xj.yj,jj+zjj.
But still the right-hand side does not give as much information as the
left-hand side ; for xj . yj .ji is not equivalent to xy ,j>
Hence the distributive power of addition in reference to multiplication to
some extent has been lost. It cannot be employed in this instance without
some loss of existential information.
(5) The symbol (x + y + co) will be taken to mean that to operates on x + y,
and therefore that x + y is not «. Thus x-^y + & implies x + a> and y + co',
but the converse does not hold. The mode of attachment of to to the term on
which it operates has some analogy to addition as it obtains in this algebra.
Thus
(a? + Q>) + (y + oo) + ft> = flj + y + G),
though (a? + o)) + (y + 0)) is not equivalent to (a; + y + a>) as far as its exist-
ential information is concerned.
Again, if x, y,ZyUj.,. denote any number of regions, then
(x + y-{-z + u+...+(o)=[(x+a))+(y + (a)'\-(z-\-<a)+... + <o};
but the final (o cannot be omitted if the existential information is to be the
same on both sides.
The distributive law of addition in relation to multiplication (cf. § 24,
equation B) does not hold completely.
Consider the expression xy + a>. Now xy can only be i, if both x and y
are equivalent to i.
Hence xy+ a> implies either x + a) or y + o) or both. Thus we may
adapt the symbolism so as to write
a;y + © = (a? + c«>i) (y + o)i) ;
where the suffixes of the o>'s weaken the meaning to this extent, that one
of the fi)*s is to hold good but not necessarily both. We define therefore
(x + «i) (y -h a>i) (z + ©i). . . to mean that one at least of the terms x,y,z,,.. is
not i.
86 EXISTENTIAL EXPRESSIONS. [CHAP. III.
It is obvious that (a? + wj + cd) (y + <»i) {z + aoi) ... =(x + a>)yz,,.y
since a? + ©i + o) ensures definitely that a; is not i.
For example,
{x + (o) y + (o = {x + €0 + (Oj) {y + (Oi) = (x + (o) y.
Let the symbols such as ji or Wi be called weak symbols in contrast to j
or (k> which are strong symbols. Then a strong sjonbol absorbs a weak
sjm[ibol of the same name (J or o)) when they both operate on the same term,
and destroys all the companion weak symbols. Thus
^jd + yji = ^j + y» (a? + «! + «) (y + <»i) = (x-^-a>) y.
(6) The chief use of this notation arises from its adaptation to the
ordinary transformations owing to the following consideration. If x exists,
then X cannot be i ; and conversely if x be not i, then x exists.
Hence ~ {xj) = ^ + &>, and " (^ + w) = xj.
But by analogy to § 26, Prop. VI.
- {a^ = X + J, and " (a + o)) = xo>.
Hence we may write J = «, and to =jy corresponding to iJ = 0, and 0 = %.
Thus the original existential information can be retained through any
transformations of the algebra.
40. Umbral Letters. (1) This existential notation can be extended.
Let the letters of the Qreek alphabet be taken to correspond to the letters of
the Roman alphabet, so that a corresponds to a, fi to b, and so on.
Let xa mean that the regions x and a overlap ; in other words xa implies
xa .j, but the symbol xa in itself denotes only the region a?; it only implies
this extra information. Also let d;+a, while denoting only the region x,
imply that x does not include all the region a; in other words x + a implies
xa.j, that is, it implies x + a + <o. Thus xa implies aj and xj and xa.j;
while x + a implies a;, ^^' and xa.j. Also xa does not necessarily exclude
xa, and a? + i does not necessarily exclude x + a.
(2) Now if X includes some of a, it follows that x cannot include all a.
Hence if xa, then x + a. This can be expressed by the equation
- (xa) =x + a.
Thus for instance, ~ (xa) .y = (x-{-a)y.
Also it follows that " (xa) = ~" (^ + a) ;
and hence " (x-\-a)=xa.
But by analogy to § 26, Prop. VI.
"" (x + d) = ^5 = xd.
40] UMBRAL LETTERS. 87
Hence we may write 1 = a ; though as a matter of fact the Greek letters
have no meaning apart from the Roman letters to which they assign
properties, and therefore should not be written alone.
(3) Let these Greek letters be called shadows or umbral letters ; and
let the Roman lettei*s denoting regions be called regional lettera
Then the umbral letters essentially refer to some regional letters or
groups of letters and are never to be separated from them. Thus a(b + y)
cannot be transformed into ab + cuy; the symbol (6 + 7) is essentially one
whole, and the bracket can never be broken. Similarly a . by cannot be
transformed into ab .y; since 67 is one indivisible symbol.
But with this limitation — that brackets connecting regional and umbral
letters are never to be broken — it will be found that the umbral letters
follow all the laws of transformation of regional letters.
(4) In accordance with our previous definitions it may be noted that
x{a-^fi) implies x(a + b) .j, and (x-^a + fi) implies that x does not include
all - (a + 5).
Also xafi implies axib.j, and {x + afi) implies that x does not include
all ~{pb\ that is, all (a + 6).
It is further to be remarked that x{a-\-P) is not identical in meaning
with xa + xp. For a? (a + )8) implies a? (a + 6) . j, that is either xa .j or xb ,j
or both, while xa + xfi implies both xa,j and ab .j.
Now xafi implies xab.j, that is both xa,j and xb.j as well Bsxab.j.
Hence xeifi implies all that xa + xfi implies and more, and xoL + xfi implies
all that x{a-{'fi) implies and more; while all three expressions represent
the same region, namely x,
(5) The shadows follow among themselves all the symbolic laws of
ordinary letters.
For a;(a +i8) = x{fi + a), a? + (a + )8) == a? + (^8 + a),
xafi = a?/8a, x + a0 = x + fia,
a:(a+ a)=aw, a7 4-(aH-a) = fl; + a,
xa{fi-hy) = x (afi + a7), x+a{0 + y)^x + (afi + ay),
x'(al3) = x{a + jS), x + -{afi) = x + (a + )3),
a? - (a + /8) = xa^, x + -(a+fi) = x + afi.
Apart from this detailed consideration it is obvious that the same laws
must hold ; for the shadows also represent regions, though these shadowed
regions are only mentioned in the equations for the sake of indicating
properties of other regions in reference to them.
It should also be noticed that since xafi implies xab.j, it also implies
ab ,j.
88 EXISTENTIAL EXPRESSIONS. [CHAP. IIL
Other transformations are
- {a;(a + /8)) = flc + -(a + )8) = ^ + aiS,
-{a? + (a + i8)}=^-(a+)8) = ^a^,
- {xa^} = x-\--'(aj3) = x + (a+l3\
^ {a? + - (a/3)} = xaj3,
-{a?-(a + i8)} = S + (a + /3).
It is to be noted that with the symbol a? (a + /8), we may not transform
to a?a 4- xfi, and thence infer fl?a and xfi ; the true transformation is
a?(a + /8) = fl?ai + a?A,
where ai and fii are weak forms of a and 13,
Similarly we may not transform a? + (a + )8) into {x + a) + {x + fi) and
thence infer a? + a and x + fi.
(6) Each complex umbral symbol should be treated as one whole as &r
as symbolic transformations are concerned. Thus the laws of unity and
simplicity (cf. § 25) have to be partially suspended. For instance xa + xfi
denotes only the region x, but for the purposes of the existential shadow
letters xa and xfi must be treated as distinct symbols. Similarly xa.xjS
denotes only the region x, but it does not mean the same as xafi ; for a;a . xfi
denotes the region x and implies xa,j and (cb.j, whereas xafi denotes the
region x and implies xab .j. The second implication includes the first, but
not the first the second. Hence for the purposes of multiplication xa and
xfi must be treated as different symbols. The suspension of these laws of
unity and simplicity causes no confusion, for the symbols are only to be
treated as different symbols (although denoting the same region) when they
are so obviously to the eye ; thus xa and x0 are obviously different symbols.
(7) When the same regional letter is combined with various umbral
letters, the same result is obtained whether the expressions are added or
multiplied*.
Thus
a?a + a?/8 = aw . x/3,
xa + fi =(x + l3)a.
(8) This notation enables existential expressions to be transformed.
Thus if f corresponds to a?, 17 to y, and ^to z,
an/.j = ayn.y^.
Hence xy .j + z = (xr^ + z) (y^ + z);
and in this case the connotation is exactly the same on both sides. Hence
the distributive power of addition in reference to multiplication has now
* This remark is dae to Mr W. E. Johnson.
41] ELIMINATION. 89
been retained. It may be noticed that the right-hand side might have been
written (xTf + z)(y + z) without alteration of connotation ; for anf implies aj,
yjy ^ 'jy c^d the { affixed to y implies no more.
Again, «? + y + « = (a? + 17) + (y + f ),
where {x + v) implies that x does not include all y and y + f implies that y
does not include all x.
Thus (x+y'\'a>)z = (x + r})Z'\-{y + ^)z,
the connotation of both sides is the same. Thus the distributing power of
multiplication in reference to addition has now been retained.
It is to be noticed that symbols like x + t) and xfj are to be treated as
indivisible wholes.
Again as examples consider the transformations
'-(xy.j) = -(xtf.yS) = (x + fl) + (y + ^);
and "(« + y + «) = -|(a? + i7) + (y + f)} = ^.yf
41. Elimination. (1) It is in general possible to eliminate x,y,z,...
from existential expressions of the forms
/(«, y, «, ... t)j and f{x, y, z, ... t) + ©.
Consider first the form f{x, y, z,..,t)j.
Let f{x, y, z, ...t) be developed and take the form
axyz ...t-^bxyz ... i-\- ,..+g'xyz ... i.
By § 33 (2) the maximum extension of the field of this expression is
Hence if /(a?, y, ^, ... t)j, the maximum extension cannot be null. Thus
(a + b + ... -\-g)j
is the resultant expression when x, y, z, ... t have been eliminated.
(2) Consider the form, /(a?, y, z,...t) + <o.
This is equivalent to / (a?, y, z, ...t) .j.
If f(x, y, z,...t) be developed as in (1), then the existential expression
becomes
(axyz ... 1+ bxyz ... i+ ,.. -\-gxyz ... i)j.
Hence by (1) (a + 6 + . . . + g)j,
that is ab ... g + 09.
This result might also have been deduced by noticing that db ... g is the
minimum extension of the field o(f{x, y, z, ...t); and therefore is necessarily
not i, i{ f{x, y, z, ... t) is not i.
90 EXISTENTIAL EXPRESSIONS. [CHAP. III.
(3) Ajs particular cases of the above two subsections, note that
{ax-{-bx).j yields (a + 6)j,
{au 4- hv) .j yields (a + b)j,
(ax + bx) + a> yields ab + a>.
Also note that (au 4- 6t;) + © yields no information respecting a and b ; for
when the formula of (2) is applied to its developed form the resultant becomes
0 + (o, which is an identity.
(4) To eliminate x,y,z,...t from f(x, y, z,...t) j and from n equations
involving them.
Let f(x, y, z, ,..t) be developed as in (1), and let the corresponding
resultant discriminants of the equations h^ A, B, C, ,,. 0,
Then the maximum extension of the field o{f(x, y, z, ...t) as conditioned
by the equations is ail + 65 4- ... + gG.
Now f(x, y, z ...t)j requires that the maximum extension shall not be
null. Hence the complete existential expression* to be found by elimina-
tion is
(aA +bB+...+gG)j.
Let this be called the existential resultant.
The resultant found by elimination o{ x, y, z,... t from the equations is
AB...Q--0.
The existential resultant and the resultant of the equations contain the
complete information to be obtained from the given premises after the
elimination of x, y, ,.. t
(5) An allied problem to that of the previous subsection is to find the
condition that the existential expression may not condition x, y, z, ...t any
further than they are already conditioned by the equations.
The minimum extension of the field of /(a?, y, z,..,t) as conditioned by
the equations is by § 33 (8),
(a + A)(b + B)...(g + G).
Hence if (a + 2)(6 + B)...(g+G)j,
then f(x, yt Zy... t)j, for all values o{ x, y, z, ...t; and thus f(x, y,2,.., t)j
does not condition x,y,z,... t
The condition can also be written
(ail +bB+...gG + a)).
(6) A special case of (5) arises when there are no equations ; the exist-
ential expression does not condition the unknowns, if
ahc "• g ' j'
* This expression found by another method was pointed out to me by Mr W. E. Johnson.
42] SOLUTIONS OF EXISTENTIAL EXPRESSIONS WITH ONE UNKNOWN. 91
(7) If the existential expression be f{x, y, z, ...ty-^-o), then by reasoning
similar to that in subsections (4) and (5) the existential resultant is
(a + A)(b + B)...(g-hO)-^(a.
The condition that the unknowns are not conditioned by the existential
expression is
(aA +bB-\- ... +gO + (o).
These conditions may respectively be written
(aA + bB+ .,. +gG)j.
and (a -h A)(b + B) ... (g + 0) j.
42. Solutions of Existential Expressions with one unknown.
(1) Solution of ax.j. The form of solution for x can be written in two
alternative forms by using symbols for undetermined regions : thus
x^iva.j + iiaB pou
The first form states explicitly that x is some (not none) undetermined
part of the region a together with some (or none) of a. The second form
states the same solution more concisely but perhaps less in detail : it states
that X may be any region p, so long as j> is assumed to include some (not none)
of the region a. There is no reason in future to write p for the undeter-
mined region denoted by x. Thus we shall say that the solution of dx .j is
x = xa,
(2) Solution ofbx.j. From the preceding proposition
x^wb.j + u = xfi.
Hence x = ~ (wb .j-\-u) = ~' (x^)
= u(w + b + oi)) = x-\-^.
The form u{w+b + oa) states that x must be some (or none) of a region
which is composed of all b and of any other region, except that the total
region must not comprise all the Universe. The form x + 0 states that x
may be any region so long as it does not comprise all b.
(3) Solution of any number of expressions ax .j, a'x.j, ... a^x.j.
The required solution is obviously
x = xa + xa! + ... + xa^
= Xxa (say).
It may be noticed that x{a + a' + ... +a**) is not the required solution,
since it is only equivalent to the weakened form a?ai + ^ai'+ ... + a?ai**; also
that a:wa'...'a** implies a^\..a^.j and xa^' ... a^ .j, which is more than
is given by the equations.
By § 40 (7) the solution can also be written
x = xa.xa' ... xa^ = 11 (xa) (say).
92 EXISTENTIAL EXPRESSIONS. [CHAP. lU.
(4) Solution of any number of expressions of the types
bx,j\ Vx.j, ... b^x.j.
The required solution is
= X(x + fi)(&a.j),
(5) Solution of any number of expressions of the types
ax.j, a'x.j, ... a^x.j, bx,j, Vx.j,.,. b^x.j.
The solution is obviously
x = Xt(x + l3)a.
If there are only two such expressions, namely aa.j and bx.jy the
solution becomes
X = (X+^)CL
(6) Solution of {ax + bx) . j.
Now (cKP 4- bx)j = a4V.ji + bx .j\.
By subsection (5) ax .ji and bx .ji imply
x = (x + fii)ai;
where ai and fii are alternative weakened forms of shadows.
But this expression does not necessarily imply any restriction on x. For
ax-^-bx can only vanish if a6 = 0.
Hence {ax + bx)j either implies ab.j and x entirely unconditioned, or
ab = 0 and
x = (x + fii)ai.
(7) Solution ofax + to. Now aa + m implies ~ (cm? 4- ©), that is (a + x)j.
But {a-\'X)j = (ax + x)j. This implies either aj and x entirely un-
conditioned, or a = 0, that is a = i, and x + ta.
(8) Solution of bx-\-{o.
Now hx-Vto implies ~ (Jbx + o)), that is (6 + a?)j.
But (b-\-x)j^Q>x-\'x)j. This implies either bj and x entirely uncon-
ditioned, or 6 = 0, that is 6 = i, and aj.
(9) Solution ofa^x + l^ + to.
Now aa) + bx+ a> implies "* (ax + bx + a>).
But ~(aa? + ^ + fi>) = (ax'\-bx)j.
Hence either a6 . j (that is, a 4- 6 4- «) and x is entirely unconditioned, or
a6 = 0 and x^(x+ ySi) di ;
where ai and y3i are weak forms.
43] EXISTENTIAL EXPRESSIONS WITH TWO UNKNOWNS. 93
43. Existential Expressions with two unknowns. (1) The general
fonn of tlie existential expression involving two unknowns, x and y, is
(axy + ha^ + c^ + d^) j.
Let /(a?, y) stand for the expression axy + hay + cxy + d^.
If ahcd . j, the above existential expression does not condition x and y
in any way (of. § 41 (6)).
But if ahcd^O, then /(a?, y) vanishes (cf. § 34 (5)), if
x = cd + u(a + b)y y^bd + v(a + c) (1);
where u and v satisfy the unlimiting equation
abcuv + abduv + acduv + bcduv = 0.
Thus if /{x, y) is to vanish the minimum extension of the field of a; is cd,
its maximum extension is a + 6, the minimum extension of the field of y is
bdy its maximum extension is a + c.
Accordingly,/(aj, y)j and a6cd = 0, yield three cases:
(a) X lies outside its above-mentioned field, and y is unrestricted :
()3) y lies outside its above-mentioned field, and x is unrestricted :
(7) both X and y lie within their respective fields, but do not occupy
mmdtaneous positions within their fields. That is to say, x and y can both be
expressed by equations (1), but {abcuv + abduv + acduv + bcduv) j.
If /(a?, y) = 0 be an unlimiting equation for x and y, then cases (a) and
(J3) necessarily cannot be realized ; and the existential expression in case (7)
becomes /(t^, v) j, where u and v are written instead of a? and y.
Case (a) is symbolized by a?= {^ + ~(xS)i} (a^SX, where x ^ *^® umbral
letter of 0 and the suflSxes denote alternative weak forms. This existential
expression for x implies that either x does not include all cd, or x does include
some region not (5 + 6).
Case (13) is symbolized by y = {y + "" (l3S)i] (ax)i-
Case (7) requires that the problem of the next subsection be first con-
sidered.
(2) To solve for x and y from the expression, /(a?, y)j; where /(a?, y) = 0
is an unlimiting equation.
No expression for x or for y can be given, which taken by itself will
satisfy /(a?, y)j : for since the equation, /(a?, y) = 0, is unlimiting any value of
a? or of y is consistent with its satisfaction. Thus to secure the satisfaction of
f{x, y)j, either xor y must be assumed to have been assigned and then the
suitable expression for the other (i.e. y or x) can be given. Thus write
fip* y) 3 = {(cty -\'by)x+ (cy + dy) x} j.
Then by § 42 (6), if y be conceived as given,
^ = {^ + (X^ + ^v)i} (av + ISv)i'
94 EXISTENTIAL EXPRESSIONS. [CHAP. III.
Similarly if x be conceived as given
Both these expressions for x and y hold concurrently, and either of them
expresses the full solution of the problem.
(3) Returning to the general problem of the solution of
{axy -^bxy + cxy + dxy)j\
where abed = 0 ; the different cases can be symbolized thus :
(a) x={x + '(xS),}{al3),.
(0) y = {y+-()9S)i}(ax)i>
(7) x={x + (xv + S^Xl {<^V + fiv)i»
or y={y + ()8f+g|),}(af + xlX
where x and y have the forms assigned io equations (I) of subsection (1).
44. Equations and Existential Expressions wrrn one unknown.
(1) Let there be n equations of the tjrpe ayX + brX = CrX + d^x ;
and an existential expression of the type ex . j.
Let A and B be the resultant discriminants of the n equations. Then
the total amount of information to be got from the equations alone is
(cf.§30), __ _ .
AB — Q, and x^B+uA,
The full information to be obtained by eliminating x is (cf. § 41 (4)),
AB^O, eA .j.
In considering the effect of the existential proposition on the solution
for X two cases arise. For x = B + uA, where u is conditioned by
e{B + uA).j.
Hence either (1) eB,j\ x^Be-k-uA, in which case u is entirely uncon-
ditioned (cf. § 41 (5)); or (2) eB — 0, and ueA .j.
If the coefficients such as e, Or, b^ etc. be supposed to be known, then
any result not conditioning u may be supposed to give no fresh information.
Thus in case (1), where eB . j, this result must be supposed to have been
previously known, and therefore the existential expression ex . j adds nothing
to the equations. But in the case (2), v>eA . j gives u = iiea, where a is the
umbral letter of A. Hence the solution for x is
x = B + uA . ea.
Here the existential expression ex.j has partially conditioned u, and
thus has given fresh information.
44] EQUATIONS AND EXISTENTIAL EXPRESSIONS WITH ONE UNKNOWN. 95
(2) Let there be n equations of the type OrX + hrX == CrOc + dtX,
and an existential expression of the type ^ .j.
The resultant of the equations is il ^ = 0, and their solution is
x = B + uA,
Hence x = A-\- uB.
Hence e (il + uB) . j.
The resultants -45 = 0, eB,j contain the full information to be found
by eliminating x (cf. § 41 (4)).
The solution for x falls into two cases; either (1) eA .j, and u is not
conditioned (cf. § 41 (5)) ; or (2) eA = 0, and ueB . j.
If the coefficients be assumed to be known apart from these given
equations, then the solution in case (1) must be taken to mean that the
existential expression adds nothing to the determination of x beyond the
information already contained in the equations. But in case (2) u is
partially determined; for from HeB.j, we deduce u = U€fi, where )8 is the
umbral letter of B.
Hence Ur= (u + €+ 13).
Therefore if e-4 = 0,
x = B + {U'{-€ + ^)A.
In this case the existential expression has given fresh information.
(3) Let there be n equations of the type OfX + b^x = CrX + dfX,
and an existential expression of the type {ex + gx)j.
The resultant of the equations is AB = 0, and their solution is
x^B + uAy x = A + uB.
Hence {eB + gA + eAu'\- gBu] j.
The resultants AB=^0, (eA-\-gB)j contain the full information to be
found by eliminating x (cf. § 41 (4)).
The solution for w falls into two cases, according as the existential
expression {eB + gA + eAu-^ gBu}j does not or does condition u.
Case (1). If (eB-^gA -^egAB)j, then the above existential expression
does not condition u at all (cf. § 41 (5)).
Hence if the coefficients are assumed to be known apart from the in-
formation of the given equations and existential expression, then the exist-
ential expression must be considered as included in the equations.
Case (2). If {eB + gA +egAB)=^0, then the existential expression for u
reduces to (eAu + gBu)j, where egAB^O,
Hence (cf. § 42 (5)) the solution for u is
96 EXISTENTIAL EXPRESSIONS [CHAP. III.
where the 'suffix 1 to the brackets of the umbral letters implies that they are
alternative weak forms.
Hence the solution for a; is in this case
In this case the given existential expression is to be considered as giving
fresh information.
45. Boole's General Problem. (1) This problem (c£ § 33 (8)) can
be adapted to the case when existential expressions are given, as in the
following special case.
Let there be given n equations of the type OfO! + tr^ = Cr« + dfX,
and an existential expression of the type gas.j; it is required to determine z,
where z is given by
z = ex+/x.
By § 33 (8), z = eB-^/A +veA + ufB,
where x=^B-\-uA,
Hence from above either (1) gB.j, and u is unconditioned by the
existential expression, or (2) gB = 0, gAu .j. In the second case ti = urya.
Hence if gB . j the existential expression adds nothing to the solution,
assuming that the coefficients are already known ; it gB = 0, then
z = eB-¥fA + eA. uya +fB (u + 7 + a).
It is to be noticed that even in the second case the existential expression
gives no positive information as to z, and that it only suggests a possibiUty.
For the solution asserts that u contains some of gA, but eA need not overlap
that part of gA contained in u. Similarly the umbral letters in the ex-
pression/B (16+7 +5) give no definite information as to the nature of
the term.
(2) If the existential expression in this problem be of the type gx .j, then
if gA ,j, it is included in the equations. But if gA = 0, the solution for
z is
z = eS +fA + eA (u + ^ +y)+fB .u/Sy.
Similar remarks apply to this solution as apply to that of the previous
form of the problem.
(3) If the existential expression be of the type gz ,j or gz .j, then more
definite information can be extracted. Take the first case, namely gz,j,
as an example.
The solution for z from the equations is (cf. § 33 (8))
z = (ef+ eB +fA) + u {eA +fBl
where e/+ eB +/A 4 eA +/B.
46] EQUATIONS AND EXISTENTIAL PROPOSITIONS WITH MANY UNKNOWNS. 97
The existential expression requires the condition
If the coefficients are assumed to be well-known, then if
g(ef+eB-\-fA),j,
no information is added by the existential expression. But if
g{ef+eB+fA)^0,
then z = («/+ e B +fA ) + {eA+fB)u (76a + y(f>0\
where ^ is the umbral letter off.
The solution for gz .j is similar in type.
46, Equations and Existential Propositions with many unknowns.
(1) A more complicated series of problems is arrived at by considering the
set of n equations involving two unknowns of the type
OfOy + bfOiy + Cr^ + d^ = Or'a^y + Waiy + c/«y 4- drXp (1) ;
combined with the existential expression of the type
{exy -^-fxy +gxy -^Wy) ,j (2).
The various discriminants of the t}rpical equation are
Also the resultant discriminants are
A^n{Ar\ B = U(Br\ C-n(a), D^U{Dr).
Then from § 30 (9) the resultant of the equations is ABCD = 0,
and from § 41 (4) the existential resultant is (eA +fB +gC+lD).j.
If (e-\-A)(f+B){g^C)(l + D),j,
then by § 41 (5) the existential expression (2) adds nothing to the equations (1)
as regards the determination of x, assuming that the coefficients are well-
known.
Assume that (e + A)(f+ B) (g + C) (Z-l- 5) = 0.
The solutions of the equations for x and y can be written
x = (A+B)u'\- CDu, y = (A-\-C)v+ BDv,
where u and v satisfy
ABGvv + ABDuv-\-ACDuv + BCDuv = 0 (3).
Substituting in (2) for x and y,
[[e(A +BC)+fABC+gABC-¥lASG] uv
+ {eABD+f(B+AD)+gABD-^lABD]uv
+ {eACD-^fACD+g{G + AD) + lACD}uv
-\-{eBGl) + fBGD+gBCD + l(D + BG)}uv],j (4).
W, 7
98 EXISTENTIAL EXPRESSIONS. [CHAP. III.
The equation (3) is unlimiting and the problem now becomes that of the
next subsection.
(2) Given an unlimiting equation (5) and an existential expression
{exy +fjcy +gxy + Ixy) . j (6)
to find the solution for x and jy.
Let A, B,G, Dhe the discriminants of the equation (5). Then, as before,
the condition that x and y are conditioned by (6) is
(e + A){f+B)(g^G)(l + D)^0,
Since the equation (5) is unlimiting, this equation can be written
efgl +fglA + gleB + lefC+ efgD = 0.
Let a symmetrical solution of the equation (5) according to the method of
§35 be
X = diuv + hiuv 4- Ciuv 4- d^uv,
y = Otuv + h^uv + CjUt; + d^uv.
Let the expression (6) be written /(a?, y).j for brevity.
Then substituting in (6) for x and ^, as in § 33 (2), the expression becomes
{/(«!, Oj) uv +f(bi, 6,) uv +f((h, c,) uo +f(di, dj) uv] .j.
But this expression has been solved in § 43.
NoTB. — In this discussion of Existential Expressions valuable hints have been taken
from the admirable paper, * On the Algebra of Logic/ by Miss Christine Ladd (Mrs Franklin)
in the book entitled Stvdies in Deductive Logic^ by Members of the Johns Hopkins University
But Mrs Franklin's calculus does not conform to the algebraic type considered in this
book ; and the discussion of Existential Expressions given here will, it is believed, be
found to have been developed on lines essentially different to the discussion in that paper.
CHAPTER IV.
Application to Logic.
47. Propositions. (1) It remains to notice the application of this
algebra to Formal Logic, conceived as the Art of Deductive Reasoning. It
seems obvious that a calculus — beyond its suggestiveness — can add nothing
to the theory of Reasoning. For the use of a calculus is after all nothing
but a way of avoiding reasoning by the help of the manipulation of symbols.
(2) The four traditional forms of proposition of Deductive Logic are
Allais6 (A),
Noaisft (E),
Some a is 6 (I),
Some a is not h (0).
Proposition A can be conceived as stating that the region of a's is included
within that of 6's, the regions of space being correlated to classes of things.
It is unnecessary to enquire here whether this is a satisfactory mode of
stating the proposition for the purpose of explaining the theory of judgment:
it is sufficient that it is a mode of expressing what the proposition expresses.
(3) Accordingly in the notation of the Algebra of Symbolic Logic
proposition A can be represented by
a^b (A),
where a symbolizes the class of things each a, and ( the class of things each h.
By § 26, Prop, viii, and § 28, this proposition can be put into many equi-
valent symbolic forms, namely a = ah^ 6 = a + 6.
Also into other forms involving i, a and 6; namely,
h ^ a, ab = Oy a = a + 6, a + 6=i, 6=6t6.
Also into other forms involving the mention of an undetermined class u ;
namely
a^vby h — a-\-Uy a = b + u.
7—2
100 APPUCATION TO LOGIC. [CHAP. IV.
(4) According to this interpretation i must symbolize that limited class
of things which is the special subject of discourse on any particular occasion.
Such a class was called by De Morgan, the Universe of Discourse. Hence
the name, Universe, which has been adopted for it here.
(5) Proposition E can be construed as denying that the regions of a's
and Vb overlap. Its symbolic expression is therefore
a6 = 0 (E).
This can be converted into the alternative forms
a^b, b^a, a = ab, b = bd, a4-6 = i, a = a + b, 6 = 6 4- a.
Thus, allowing the introduction of t, there are eight equivalent symbolic
forms of the universal negative proposition, as well as eight forms of the
universal affirmative. But if the introduction of i be not allowed, there is
but one form, namely, a6 = 0 ; remembering that the supplement of a term
by its definition [cf § 23 (8)] implies i.
On the other hand if the introduction of an undefined class symbol {u)
be allowed, then four other forms appear, namely,
a = vb, a = it + 6, 6 = via, 6 = il + a.
(6) Proposition I can be construed as affirming that the regions of the
a's and 6's overlap. Hence it affirms that the region ab exists. This is
symbolically asserted by
abj (I).
Equivalent forms are (cf § 40) a/8 . 6a; a + 6 + oi; (a 4- )8) + (6 + a).
Also if the introduction of undefined class symbols be allowed, then other
equivalent forms are,
a = wb.j + u', 6 = wa,j + w; a = n(7Z; + 6 + ©); b = u(w + a-hw).
(7) Proposition O affirms that the regions of a's and 6's overlap. This
is symbolically expressed by
ab.j (O).
Equivalent forms are a/8 . 6a; a + 6 + fi>; (a + )8) -f (6 4- v).
Also using undefined class 83nnbols,
a = wb.j + u] b = u{w + a + a>y, a - li (w + b -h (o); b = wa.j +u.
48. Exclusion of Nugatory Forms. (I) It is sometimes necessary
to symbolize propositions of the type A, so as to exclude nugatory forms ;
for instance when it is desired to infer symbolically a particular proposition
from two universals.
(2) In order to avoid the form of nugatoriness which would arise from
a = 0, in a 4 ^» we can write
cy4^* (1).
or a; = a6.^* (2).
49] SYLLOGISM. 101
The series of other forms can be deduced by mere symbolical reasoning
from this form. Thus b^b + ab; also bj, ah .j, and ab ,j = aj; hence
bj = bj + aj (3).
Again, by taking the supplement of bj, we deduce 6 + ©. Multiplying (2)
by (6 4- ft>), we find
aj.(b + w) = 0 (4).
By taking supplements of (1),
b + uf^a + a (5).
By taking supplements of (2)
a + a) = a + 6-fG) (6).
By taking supplements of (3)
6-f o> = (6H-w)(aH-o>) (7).
By taking supplements of (4)
{a-ha>) + lj = i (8).
Thus the eight forms of the proposition (A) (excluding those with un-
determined class sjnnbols) have been symbolized so as to exclude the nugatory
form which arises when a = 0.
(3) Another nugatory foini arises when b = i, this form can be excluded
by the forms
a + a)^b + (o, or (a 4- «) = (a 4- •») (6 4- fi>).
By comparing these forms with equations (5) and (7) in subsection (2)
it is easy to write down the remaining six forms.
It is also possible to combine the symbolism of both cases and thus to
exclude both forms of nugatoriness, viz. a = 0, or 6 = i. But it is rarely that
reasoning requires both forms to be excluded simultaneously, so there is no
gain in the additional complication of the symbolism.
49. Syllogism. (1) The various figures of the traditional syllogisms
are as follows, where a is the minor term, b the middle term and c the major
term:
First Figure.
A, All b is c, E, No b is c, A, All 6 is c, E, No b is c,
A, All a is b, A, All a is b, I, Some a is 6, I, Some a is 6,
therefore therefore therefore therefore
A, All awe. E, No a is c. I, Some a is c. 0, Some a is not c.
Second FHgwe.
E, No c is 6, A, All c is 6, E, No c is 6, A, All c is 6,
A, All a is 6, E, No a is 6, I, Some a is 6, 0, Some a is not 6,
therefore therefore therefore therefore
E, No a is <?. E, No. a is c. O, Some a is not c. 0, Some a is not a
102
APPLICATION Ho L<:)G1C.
[chap. IV.
A, All b is Cy
A, All 6 is a,
therefore
I, Some a is c.
Third Figure.
E, No 6 is c, I, Some b is c,
A| All b is a,
therefore
0, Some a is not c.
O, Some b is not c,
A, All b is a,
therefore
O, Some a is not c.
A, All 6-is a,
therefore
I, Some a is c,
E, No 6 is c,
I, Some 6 is a,
therefore
0, Some a is not c.
A, All 6 is c,
I, Some 6 is a,
therefore
1, Some a is c.
A, All c is 6,
A, All b is a,
therefore
I, Some a is c.
E, No c is 6,
A, All 6 is a,
therefore
O, Some a is not o.
or thus :
or thus :
or thus :
Fourth Figure.
A, All c is by I, Some c is 6,
E, No b is a, A, All b is a,
therefore therefore
0, Some a is not c. I, Some a is c.
E, No c is 6,
I, Some 6 is a,
therefore
0, Some a is not c.
(2) The first mood of the first figure can be symbolized thus :
6 ^ c, a 4 ^» therefore a^c:
b = bCy a =■ ah, therefore a — ah^^ abc = ac :
be = 0, a6 = 0, therefore ac = a(5 + 6)c = a.6c + a6.c = 0:
c^b, b^a, therefore c^a:
or thus : c = 6+c, 5 = a + 6, therefore c = b+c = a + b + c=^a-^c:
or thus : 6 = 6 + c, a = d + 6, therefore a = a + t = a+6 + c = d4-c:
or thus : 6 H- c = i, d + 6 = i, therefore d+c = d + bb + c
= (d+6 + c)(dH-6 + c) = i:
or thus : c = 6c, 6 = d6, therefore c = bc = cU)c = dc.
One half of these forms can be deduced from the other half by taking
supplements.
In each case the two premises, which are each of the type A, have been
written down in the same form. By combining two difiFerent methods of
exhibiting symbolically propositions of the type A many other methods of
conducting the reasoning symbolically can be deduced. It is unnecessary to
state them hera
(3) The second mood of the first figure can be symbolized thus :
fee = 0, a = ab, therefore ac = abc = 0 :
or thus : b^c, a^b, therefore a^c:
c^b, b^dj therefore c^d:
b = bc, a = ah, therefore a = ahc = dc :
c = cb, 6 = d6, therefore c^cb = cdb = cd:
or thus
or thus
or thus
50] SYMBOLIC EQUIVALENTS OF SYLLOGISMS. 103
or thus
6 4- c = i, at = 0, therefore a = a (6 + c) = a6 + ac =
ac
or thus : 6 = 6 4- c, a6 = 0, therefore a (6 -i- c) = ac = a6 = 0 :
or thus : c = c + 6, a = a6, therefore ac = a(c+6) = ac + a=a.
Eight forms have been given here but many others could be added by
corabiniug otherwise the modes of symbolizing propositions of the type A
and £.
60. Symbolic Equivalents of Syllogisms. (1) It is better however
at once to generalize the point of view of this symbolic discussion of the
syllogism. It is evident that each syllogism is simply a problem of elimina-
tion of the middle term, and the symbolic discussions can be treated as
special cases of the general methods already developed. Also the symbolic
equivalence of all the forms of a proposition makes it indifferent which
special form of a proposition is chosen as typical.
(2) Consider the first mood of the first figure: the term b is to be
eliminated from b=^bc, a = ab.
The positive discriminant of 6 = be, is c, the negative discriminant is i.
The equation, a = ab, can be written aft + a6 = ah. The positive discri-
minant is 1 ; its negative disciiminant is a.
Hence all the information to be found by eliminating 6 is
" (ct) X - (id) = 0 ;
that is ac = 0.
(3) Consider the second mood : the term 6 is to be eliminated from
6c = 0, a = ofi.
The discriminants of the first equation are c and i; and of the second
equation are i and a.
Hence the elimination of b gives
" (ci) X " (id) = 0 ;
that is ac = 0.
It is obvious that the first and second moods of the second figure are
symbolically the same problem as this mood.
(4) The third mood of the first figure is symbolically stated thus :
6 = 6c, ab .j.
Hence eliminating 6 by § 41 (4), the existential resultant iaacj.
This is symbolically the same problem as the third and fourth moods of
the third figure, and the third of the fourth figure.
(5) The fourth mood of the first figure can be symbolized thus :
6c = 0, a6 ,j.
Hence eliminating 6 by § 41 (4), the existential resultant is ac .j.
This is symbolically the same problem as the third mood of the second
figure, the sixth of the third figure, and the fifth of the fourth figure.
104 APPLICATION TO LOGIC. [CHAP. IV.
(6) The only mood in the second figure not already discussed is the
fourth ; it can be symbolically stated thus : c6 = 0, at .j.
Hence eliminating 6 by § 41 (4), the existential resultant is dc.j,
(7) In the first mood of the third figure a particular proposition is
inferred from two universal premises. It is necessai'y therefore in order to
symbolize this mood that universal propositions as symbolically expressed
be put on the same level as particular propositions in regard to the ex-
clusion of nugatory forms. The syllogism can be symbolized thus,
b}=bc.jy bj=baj,
hence bj = be ,j = bac .j, hence ac ,j.
(8) It is immediately evident that the premises assume more than is
necessary to prove the conclusion, thus b = be, instead of &; = bcj, and db.j,
instead of bj = ai> .j, would have been sufficient. This is not a syllogism with
what is technically known as a weakened conclusion, since no stronger
conclusion of this type could have been drawn. It might be called a
syllogism with over-strong premises. The syllogism of the same type with
its premises not over-strong is the third of the first figure. Hence the
symbolic treatment of that mood would serve for this one.
(9) The second mood of the third figure can be symbolized thus,
be = 0, bj = ah.j, now ab ,j = a6 (c f c) .j = abc ,j, hence ad.j.
This is obviously a syllogism with over-strong premises, since &c = 0, oi .j,
would have been sufficient for the conclusion. The syllogism of the same
type with sufficient premises is the fourth of the first figure.
(10) The fifth mood of the third figure can be symbolized thus : bc.j,
Hence eliminating 6 by § 41 (4), the existential resultant is aid .j.
(11) The first mood of the fourth figure is symbolized thus,
cj=bc.j, bj = ab,jt hence be .j = abc,j, hence acj.
This is a syllogism with over-strong premises, the corresponding syllogism
with sufficient premises is the third of the first figure.
(12) The second mood of the fourth figure is symbolized thus,
(yj = bc,jy a6 = 0, therefore be. j= be (a + a). j=bea.j, hence ca.j.
This is a syllogism with over-strong premises ; the corresponding syllogism
with sufficient premises is the fourth of the first figure.
(13) The fourth mood of the fourth figure is symbolized thus,
6c = 0, bj = ab,jt therefore ab .j = ab(e -hc),j = abc.j, hence oc./
This is a syllogism with over-strong premises ; the corresponding syllogism
with sufficient premises is the fourth of the first figure.
51] OE19ERALIZATION OF LOGIC. 105
(14) Since the conclusion of any syllogism can be obtained from the
premises by the purely symbolic methods of this algebra, it follows that the
conclusion of any train of reasoning, valid according to the formal canons of
the traditional Deductive Logic, can also be obtained from the premises by
the use of the algebra, using purely symbolic transformations.
61. Generalization of Logic. (1) This discussion of the various
moods of Syllogism suggests^ that the processes of elimination and solution
as applied to a system of etjuations and existential expressions developed in
the preceding chapters of this Book can be construed as being a generaliza-
tion of the processes of syllogism and conversion of common Logic.
It will be seen by reference to § 47 that a universal proposition is
symbolized in the form of an equation, and a particular proposition in the
form of an existential expression. Hence the most general form of equation
may be conceived as a complex universal proposition, and a set of equations
as a set of universal propositions. Also the most general form of an exis-
tential expression is the most general form of a particular proposition, and a
set of such expressions is a set of particular propositions.
(2) The most general form of a system, entirely of universal proposi-
tions and involving one element to be determined, is given in Chapter II,
§§29,80. It is
aiX + biX = CiX + diXy
ar^ + bfX = CfX-{- dfX,
Here x is supposed to be the class to be further determined, and the
other symbols all refer to well-known classes.
Then the information wanted is found by forming n functions of the type,
Ar^dfCr+afCry and n of the type, Br^brdr-hbidn and by forming the
products A = AiA^... An, B = BiB^ . . . B^. Then x^B-^-uA] with the con-
dition that AB = 0, which is probably well-known.
(3) The essential part of this process is the formation of the two regions
A and B out of the well-defined regions involved in the system of proposi-
tions. This composition of the two discriminants is a process of rearranging
our original knowledge so as to express in a convenient form the fresh infor-
mation conveyed in the system. Formally it is a mere picking out of certain
•regions defined by the inter-relations of the known regions which are the
coefficients of the equations : but the process in practice may result in a real
addition to knowledge of the true definition of x. For instance rationality
and animality may have been the characteristics of two regions among the
» Cf. Boole, Laws of Thought, chapter IX. § 8, chapter XV.
r-/"
106 APPLICATION TO LOGIC. [CHAP. IV.
coefficients in the system ; but in A and B the common part of the regions
may only occur : then it is at once known that x only involves the ideas of
rationality aud animality in so far as it involves those of humanity — a very
real addition to knowledge, though formally it is only a question of better
arrangement as compared to the original system.
(4) The undefined nature of the information given by particular pro-
positions makes it usually desirable not to deal with such propositions in a
mass, but to sort them one by one, comparing their information with that
derived from the known system of universal propositions.
Thus let the above system of universal propositions be known, and also
the proposition of the type I, viz. ex .j.
. Then from § 41 the full information to be found by eliminating x is,
AB = 0, eA ,j ; and the solution for x is, either
(1) eB.j, x = B€-^uA, or (2) eB = 0, x=B-^uA.€ol
Now propositions including a common term x are in general accumulated
in science or elsewhere just because information concerning x is required.
Also the propositions will as far as possible connect x with thoroughly well-
known terms. If we conceived this process as performed with ideal success,
then the coefficients of x and x in the above equations and existential expres-
sion must be conceived as completely known, and no information concerning
their relations will be fresh. Hence in case (1), when eB ,j, the particular
proposition {ex .j) is included in the universal propositions; but in case (2)
the particular proposition has added fresh information.
But this sharp division between things known and things unknown is not
always present in reasoning. In such a case the universals and the particular
perform a double function, they both define more accurately the properties
of things already fairly well-known, and determine the things x which are
comparatively unknown.
' The discussion of this typical case may serve to exemplify the logical
interpretation of the problems of the previous chapters.
CHAPTER V.
Propositional Interpretation.
62. Propositional Interpretation. (1) There is another possible
mode of interpreting the Algebra of Symbolic Logic which forms another
application of the calculus to Logic.
Let any letter of the calculus represent a proposition or complex of
propositions. The propositions represented are to be either simple categorical
propositions, or complexes of such propositions of one or other of two types.
One type is the complex proposition which asserts two or more simple propo-
sitions to be conjointly true ; such a proposition asserts the truth of all its
simple components, and the proposer is prepared to maintain any one of
them. The verbal form by which such propositions are coupled together is
a mere accident: the essential point to be noticed is that the complex
proposition is conceived as the product of a set of simple propositions,
marked off from all other propositions, and set before the mind by some
device, linguistic or otherwise, in such fashion that each single proposition of
the set is stated as valid. Hence if one single proposition of the set be
disproved, the complex proposition is disproved. Let such a complex of
propositions be called a conjunctive complex.
(2) The other type of complex proposition is that which asserts that one
at least out of a group of simple propositions, somehow set before the mind,
is true. Here again the linguistic device is immaterial, the. essential point
is that the group of propositions is set before the mind with the understood
assertion that one at least is true. Let such a type of complex of propositions
be called a disjunctive complex.
(3) Furthermore we may escape the difficult (and perhaps unanswerable
or even unmeaning) question of deciding what propositions are to be regarded
as simple propositions. The simplicity which is here asserted of certain
propositions, is, so to speak, a simplicity de facto and not de jure. All that
is meant is that a simple proposition is one which as a matter of fact for the
purpose in hand is regarded, and is capable of being regarded, as a simple
108 PROPOSITIONAL INTERPRETATION. [CHAP. V.
assertion of a fact, which fact may be indefinitely complex and capable of
further analysis.
Thus a conjunctive or a disjunctive complex may each of them be
regarded as a simple proposition by directing attention to the single element
of assertion which binds together the different component propositions of a
complex of either type.
(4) To sum up: all propositions symbolized, actually or potentially, by
single letters can be regarded as simple propositions : and the only analysis
of simple propositions • is to be their analysis either into conjunctive or
disjunctive complexes of simple propositions. Also a simple proposition is a
proposition which can be regarded as containing a single element of categori-
cal assertion.
63. Equivalent Propositions. Two propositions, x and y, will be said
to be equivalent, the equivalence being expressed by a? = y, when they are
equivalent in validity. By this is meant that any motives (of those motives
which are taken account of in the pai*ticular discourse) to assent, which on
presentation to the mind induce assent to x, also necessarily induce assent
to y and converaely.
64. Symbolic Representation of Complexes. (1) Let the disjunctive
complex formed out of the component propositions a, 6, c... be symbolized
by (a + & + c ...). This symbolism is allowable since the disjunctive complex
has the properties of addition : for (1) the result of the synthesis of the
propositions is a definite unique thing of the same type as the thing
synthesized, namely another proposition : (2) the order which is conceivable
in the mental arrangement of the propositions is immaterial as far as the
equivalence of the resulting proposition is concerned: (3) the components
of a disjunctive complex may be associated in any way into disjunctive
complexes ; so that the associative law holds.
(2) Let the conjunctive complex formed out of the component proposi-
tions a, &, c... be symbolized by ahc,.,. This symbolism by the sign of
multiplication is allowable: (1) since the result of the synthesis of a number
of component propositions into a conjunctive complex is definite and unique,
being in fact another proposition which can be regarded as a simple propo-
sition; (2) since the conjunctive complex formed out of the proposition a
and the complex 6 + c is the same proposition as the disjunctive complex
formed by ah and dc ; in other words aQ>-\-c) = ah'\-(ic,
66. Identification with the Algebra of Symbolic Logic. (1) It
now remains to identify the addition and multiplication of propositions, as
here defined, with the operations of the Algebra of Symbolic Logic.
The disjunctive complex a? + a? is the same as the simple proposition x.
55] IDENTIFICATION WITH THE ALGEBRA OF SYMBOLIC LOGIC. 109
For X'\-x means either the. proposition a? or the proposition a?, and this is
nothing else than the proposition x. Hence x + x = x.
(2) The conjunctive complex obeys the associative law: for to assert a
and b and c conjointly is the same as asserting b and c conjointly and assert-
ing a conjointly with this complex assertion. Hence abc = a,bc.
(3) The conjunctive complex also obviously obeys the commutative law :
thus ahc = axib = bac.
(4) The conjunctive complex formed of a and a is the same as the
simple proposition a; hence (m = a,
(5) The nuU-elemevt of the manifold of the Algebra corresponds to the
absolute rejection of all motives for assent to a proposition, and further to
the consequent rejection of the validity of the proposition. Hence x^O,
comes to mean the rejection of x from any process of reason, or from any
act of assertion. In so far as they are thus rejected all such propositions
are equivalent. Thus if a? = 0, y = 0, then x = y = 0. Furthermore if 6 = 0,
the proposition a + & is equivalent to the proposition a alone; for the motives
of validity of b being absolutely rejected, those for the validity of a alone
remain.
Hence if ft = 0, a-^b=:a.
Again, if 6 = 0, then a6 = 0; for ab means that a and b are asserted
conjointly, and if the motives for b be rejected, then the motives for the
complex proposition are rejected.
The class of propositions to be thus absolutely rejected is best discussed
later, after the discussion of the other special element.
(6) The Universe. The other special element of the manifold is that
which has been called the Universe. Those propositions, or that class of
perhaps an indefinite number of propositions, will be severally considered
as equivalent to the Universe when their validity has acquired some special
absoluteness of assent, either conventionally (for the sake of argument), or
natumlly.
This class of propositions may be fixed by sheer convention : certain
propositions may be arbitrarily enumerated and to them may be assigned the
absolute validity which is typified by the element called the Universe. Or
some natural characteristic may be assigned as the discriminating mark of
propositions which are equivalent to the universe. For instance, propositions
which while reasoning on a given subject matter are implied in reasoning
without rising to explicit consciousness or needing explicit statement at
any stage of the argument might be equated to the Universe.
The laws of thought as stated in Logic are such propositions. Again in
a discussion between two billiard markers on a game of billiards the propo-
sition, that two of the balls were white and the third red, might be of this
110 PROPOSITION AL INTERPRETATION. [CHAP. V.
character. For billiard markers such a proposition rises to the level of a
law of thought.
Again, in legal arguments before an inferior court the judgments of the
Supreme Court of Judicature might be considered as propositions each
equivalent to the Universe.
In this interpretation the name of the Universe as applied to this
element is unfortunate: the Truism would be a better name for it. Let all
propositions equivalent to the Universe be termed self-evident.
(7) The properties assigned to the Universe (i) in relation to any
proposition x are (cf. § 23 (6) and (7))
XI = X.
The validity of any proposition equivalent to the Universe being taken
as absolute, the validity of the disjunctive complex formed of this proposition
and some other proposition x cannot be anything else but the absolute
validity of the Universe. Hence the equation x + i=:i is valid for the
present interpretation.
Again, in the conjunctive complex formed of any proposition and a
proposition equivalent to the Universe, the validity of the second proposition
being unquestioned, the validity of the whole is regulated by that of the
first proposition. Hence the equation xi=:xia also valid.
(8) This conception of a class of propositions either conventionally or
naturally of absolute validity gives rise for symbolic purposes in this chapter
to an extension of the traditional idea of the conversion of propositions.
If the Universe be narrowed down to the Laws of Thought, then all the
propositions which can be derived from any given proposition x taken in
connection with the propositions of the Universe are those propositions which
arise in the traditional theory of the conversion of propositiona Hence if we
extend the Universe of self-evident propositions either by some natural or
conventional definition, we may extend the conception of conversion to
include any proposition which can be derived from a given proposition x
taken in connection with the assigned propositions equivalent to the Universe.
Thus if % be any proposition equivalent to the Universe, xi will be
considered to be simply the proposition x in another form.
(9) The supplementary proposition, x, of the proposition x is defined
by the properties,
fla = 0, a? + ^ = t.
Whatever the propositions of the Univei-se may be, even if they are reduced
to the minimum of the Laws of Thought, the logical contradictory of x
satisfies these conditions and therefore is a form of the supplementary pro-
position. But by the aid of the propositions of the Universe there are
other more special forms into which the contradictory can be 'converted.'
56,57] SYMBOLISM OF THE TRADITIONAL PROPOSITIONS. Ill
Any such form, equivalent to the contradictory, is with equal right called
the supplement of x. Thus to the billiard markers cited above the
supplement to the proposition, the ball is red, is the proposition, the ball is
white; for one of the two must be true and they cannot both be true.
(10) It is now possible clearly to define the class, necessarily of indefinite
number, of propositions which are to be equated to the null element. This
equation must not rest merely on the empirical negative fact of the apparent
absence of motives for assent ; but on the positive fact of inconsistency with
the propositions which are equated to the Universe. If the Universe be
reduced to the Laws of Thought, then all propositions equated to null are
self-contradictory. With a more extended Universe, all propositions equated
to null are. those which contradict the fundamental assumptions of our
reasonipg. Let all propositions equated to the null-element be called self-
condemned.
(11) The hypothetical relation between two propositions x and y, namely,
If y be true then x is true, implies that the motives for assent to y are included
among those for assent to x. Hence the relation can be expressed by y 4 ^>
or by any of the equivalent equational forms of § 26, Prop. VIII. And y may
be said to be incident in x»
We have now examilied all the fundamental principles of the Algebra
of S)rmbolic Logic and shown that our present symbolism for propositions
agrees with and interprets them all. Hence the development of this
symbolism is simply the development of the Algebra which has been
already carried out.
66. Existential Expressions. The symbol x,j denotes the pro-
position X and implies that it is not self-condemned. The symbol x + co
denotes the proposition x and implies that it is not self-evident. Hence,
- (aj) = a + CD, implies that the supplement of a proposition not self-con-
demned is itself not self-evident.
Umbral letters. The symbol ict) denotes the proposition x and implies that
xy is not self-condemned : the symbol x + rj implies that x-\-y is not self-
evident (cf. § 40 (1)). The whole use of umbral letters therefore receives its
interpretation.
67. Symbolism of the Traditional Propositions. (1) This system
of interpretation, which in its main ideas is a modification of that due to
Boole^ has perhaps the best right to be called a system of Symbolic Logia
It assumes the existence of an unquestioned sphere of knowledge, and traces
generally the consequences which can be deduced from any categorical
proposition or set of categorical propositions taken in connection with this
sphere of knowledge. The former mode of interpretation, by class inclusion
1 Gf. Lawi of Thought, ohap. xi.
112 PROPOSITIONAL INTERPRETATION. [CHAP. V.
and exclusion, only applied to propositions of the subsumption type : the
present mode applies to any categorical proposition, that is to any proposition
depending on a single element of assertion. Further it can symbolize any
relation in which two such propositions can stand to each other, namely,
(1) the disjunctive relation, in either of the two forms, namely, either when
the propositions can be both true or when only one can be true (i.e. by the
forms a* + y and x+yx)\ (2) the conjunctive relation ; (3) the hypothetical
relation (i.e. by the equation y = xy),
(2) A defect of the method at first sight is that it seemingly cannot
exhibit the process of thought in a syllogism.
Thus if X and y be the two premises, and z be the conclusion, then z is
true if ay be true : hence xy=^xy.z, or xy^z are two of the forms in which
an argument from two propositions to a third can be exhibited. But this
symbolism only exhibits the fact that z has been concluded from xy, and in
no way traces the course of thought.
(3) The defect is remedied by McCoU (Proc. London Math. Soc, Vols. IX.,
X., XL, XIII.), by means of the device of analysing a proposition of one of
the traditional types, A, E, I, 0, into a relation between other propositions —
thus instead of. All A ia B, consider the propositions. It is ii. It is £; then,
All A is B, is the same thing as saying that the proposition. It is ^, is
equivalent in validity to the conjunctive complex. It is A and It is JB. Hence
if one proposition is a, and the other &, the original proposition is symbolized
by a = ah. In other words, the hypothetical relation mentioned in § 55 (11)
holds between the propositions a and b.
This analysis is certainly possible ; and it is not necessary for the
symbolism that it should be put forward as a fundamental analysis, but
merely as possible. It requires however some careful explanation in order to
understand the possible relations and transformations of such propositions as,
It is A.
68. Primitive Predication. (1) Let a proposition of the type, It
is A, be called a primitive predication. In such a proposition the subject is
not defined in the proposition itself; it is supposed to be known, either by
direct intuition, or as the result of previous discourse. In the latter case the
proposition must not be considered as an analytical deduction from previous
propositions defining the * it.' The previous discourse is simply a means of
bringing the subject before the mind : and when the subject is so brought
before the mind, the proposition is a fresh synthetic proposition. A primitive
predication necessarily implies the existence of the subject. The proposition
may be in error ; but without a subject, instead of a proposition there is a
mere exclamation.
(2) If the predicate be a possible predicate, either because it is not self-
contradictory, or further because its possibility is not inconsistent with the
58, 59] EXISTENTIAL SYMBOLS AND PRIMITIVE PREDICATION. 113
rest of knowledge, primitive predication can only be tested as to its truth or
falsehood by an act of intuition. For a primitive predication is essentially a
singular act having relation to a definite intuition ; and it is only knowledge
based on definite intuitions having concrete relations with this intuition
which can confirm or invalidate it.
The propositions taken as equivalent to the Universe in the present
symbolism must be propositions deducible from propositions relating universal
ideas or be such propositions themselves. Hence if x stand for a proposition
which is a primitive predication, then a? can only be self-condemned if the
predicate be self-contradictory or inconsistent with the propositions equivalent
to the Universe.
Also X can only itself be equivalent to the universe, if there be the
convention that during the given process of inference the ultimate subject
of every proposition shall have certain assigned attributes. Then an act of
primitive predication attributing one of these attributes to a subject is
equivalent to the Universe, that is, is self-evident.
(3) If ^ be a primitive predication, x is not a primitive predication;
it may be called a primitive negation. Thus if x stands for. It is man,
then X stands for, It is not man ; that is to say, the subject may have any
possible attribute except that of man. If x be self-condemned, then x states
that the subject may have any possible attribute; thus x^i, since it is an
obvious presupposition of all thought that a subject undefined except by the
fact of an act of intuition may have any possible attribute.
If x=^i, then ^ is a denial that the subject referred to has a certain
attribute, which by hypothesis all subjects under consideration do possess ;
hence x is self-condemned : that is, ^ = 0.
(4) A primitive negation does not necessarily occur merely as the denial
of a primitive predication. The relations of the two types of proposition
may be inverted. The fundamental proposition may be the denial that a
certain predicate is attributable to the subjects within a certain field of
thought. If this proposition, which relates universal ideas, be included
among propositions which are self-evident, then any primitive denial which
denies the certain predicate is also self-evident ; and its supplement, which is
a primitive predication, is self-condemned.
69. Existential Symbols and Primitive Predication. (1) If x
stand for a primitive predication, then xj implies that the predicate is a
possible predicate of a subject in so far as the self-evident propositions
regulate our knowledge of possibility. Now xj implies ^-f-a>; this last
expression implies that the denial of the primitive predication cannot be
deduced to be true for all possible subjects of predication by means of the
self-evident propositions. This deduction is an obvious consequence of aj.
w. 8
114 PROPOSITION AL INTERPRETATION. [CHAP. V.
(2) Also ^ implies that the denial of the primitive predication is con-
sistent with the self-evident propositions as &r as some possible subjects of
predication are concerned. Now xj implies x + <o, and this implies that the
primitive predication is not self-evident for all possible subjects of predi-
cation.
(3) If X, y, z, etc., all stand for separate primitive predications, then in
any complex, either conjunctive or disjunctive, which comprises two or more
of these propositions, the propositions are to be understood to refer to the
same subject. Otherwise, since the propositions are singular acts, the pro-
positions can have no relation to each other. Thus xy, i.e. x with y, stands
for the combined assertions, It is X and it is F, or in other words, It is both
X and F. Also x+y stands for, it is either X or F or both. Similarly
primitive denials occurring together in a complex must both refer to the
same subject; so also must primitive predications and primitive denials
occurring together in a complex.
(4) The symbol xi] stands for the proposition, It is X, and also implies
the consistency with the self-evident propositions of the proposition. It is F,
as applied to the same subject as x. The umbral letter rj affixed to a? is in
fact a reminder that xy is consistent with the self-evident propositions for
some possible subjects of predication.
60. Propositions. (1) It is now possible to symbolize the traditional
forms of logical proposition.
Proposition A. All X is F, takes the form, if x then y, where x and y
are the primitive predications, It is X, It is F Hence the proposition takes
the s}rmbolic forms
x^y,x^xy, or any symbolically equivalent form.
(2) Proposition R No Z is F, takes the form. If x then y. Hence
the proposition takes the symbolic forms
^4 y> icssaiy, or any symbolically equivalent form.
«
(3) Proposition I. Some X is F, takes the form that the conjunctive
complex xy is not self-condemned ; if the denial of all predicates or combina-
tions of predicates, which do not actually occur in subjects belonging to the
field of thought considered, be included among the self-evident propositions.
Hence the proposition can be put in the symbolic form, ay.jy or in any
symbolically equivalent form.
It must be carefully noticed that it is the connotation o{ xy.j which
expresses the Proposition I and not the conjunctive complex ay, which stands
for. It is X and F. Thus the supplement of xy . j, namely, "" (xy . j), or
x + y + co, does not express the contradictory of the Proposition I, but the
contradictory of the conjunctive complex xy. On the contrary the connotation
of S -f- ^ + « still expresses the same Proposition I.
60] PROPOSITIONS. 115
(4) Pboposition O. Some X is not Y, takes the form that the oon-
juDctive complex ity is not self-condemned ; where the same hypothesis as to
the self-evident propositions is made as in the case of Proposition I. The
symbolic form is therefore, ay .j, or any equivalent sjrmbolic form.
(5) The universal Propositions A, E as symbolized above give no
existential import to their subjects. But the symbolism as there explained
has the further serious defect that there is no symbolic mode of giving
warning of the nugatoriness of the propositions when the subject is non-
existent. But this can be easily remedied by including among the self-
evident propositions the denial of any predicates which do not appear in an
existent subject in the field of thought. This is the same supposition as had
to be made in order to symbolize / and 0. Hence in the proposition x^ocy,
if there be no X's, then a? = 0.
Also if it be desired to exclude this nugatory case, then the proposition
can be written
(6) It has now been proved that the present form of interpretation
includes that of the preceding chapter as a particular case. Thus all the
results of the previous chapter take their place as particular cases of the
interpretations of this present chapter.
Historical Note, The Algebra of Symbolic Logic, viewed as a distinct algebra, is due
to Boole, whoso 'Laws of Thought' was published in 1854. Boole does not seem in this
work to fully realize that he had discovered a system of symbols distinct from that of
ordinary algebra. In fact the idea of 'extraordinary algebras' was only then in process
of formation and he himself in this work was one of its creators. Hamilton's Lectures on
Quaternions were only published in 1853 (though his first paper on Quaternions was
published in the Philosophical Magazine, 1844), and Grassmann's Atudehnungdehre of 1844
was then imknown. The task of giving thorough consistency to Boole's ideas and
notation, with the slightest possible change, was performed by Venn in his 'Symbolic
L(^c,' (1st Ed. 1881, 2nd Ed. 1894). The non-exclusive symbolism for addition (i.a
x-\-y instead of x-\-yx) was introduced by Jevons in his 'Pure Logic,' 1864, and by
C. S. Peirce in the Prooeedingg of the American Academy of Arts and Sciences, VoL vn,
1867. Peirce continued his investigations in the American Journal of Mathematics,
Vols. in. and vil The later articles also contain the symbolism for a subsumption, and
many further symbolic investigations of logical ideas, especially in the Logic of Relatives,
which it does not enter into the plan of this treatise to describe. These investigations
of Peirce form the most important contribution to the subject of Symbolic Log^o since
Boole's work.
Peirce (loc, cit. 1867) and Schroder in his important pamphlet, Operationskreis des
LogikkalkiUsy 1877, shewed that the use of numerals, retained by Boole, was unnecessary,
and also exhibited the reciprocity between multiplication and addition ; Schr^er {J,oc, cit.)
also shewed that the operations of subtraction and division might be dispensed with.
Schroder has since written a very complete treatise on the subject, 'Vorlesungen uber
die Algebra der Logik,' Teubner, Leipsic, Vol. i, 1890, Vol. ii, 1891, Vol. ra, 1896; Vol. in.
deals with the Logic of Relatives.
8—2
116 PROPOSITIONAL INTERPBETATION. [CHAP. V.
A small book entitled ' Studies in Deductive L(^c,' Boston 1883, has in it suggestive
papers, especially one bj Miss Ladd (Mrs Franklin) ' On the Algebra of Logic,' and one
bj Dr Mitchell * On a new Algebra of Logic'
A most important investigation on the underlying principles and assumptions which
belong equally to the ordinary Formal Logic, to Symbolic Logic, and to the Logic of
Eelatives is given by Mr W. £. Johnson in three articles, ' The Logical Calculus,' in Mituiy
VoL I, New Series, 1892. His symbolism is not in general that of the Algebraic type
dealt with in this work.
The prepositional interpretation in a different form to that given in this work was
given by Boole in his book: modifications of it have been given by Venn (Symbolic
Logic), Peirce {loo, cit.\ H. M^oll in the Proceedings of the London McUkemcUioal Society,
Vols. IX, X, XI, XIII, ^ On the Calculus of Equivalent Statements.' The latter also introduces
some changes in notation and some applications to the limits of definite int^rals, which
are interesting to mathematicians.
A large part of Boole's ' Laws of Thought ' is devoted to the application of his method
to the Theory of Probability.
Both Venn and SchrOder give careful bibliographies in their works. These two
works, Johnson's articles in Hind, and of course Boole's ' Laws of Thought,' should be
the first consulted by students desirous of entering further into the subject.
There is a hostile criticism of the utility of the whole subject from a logical point of
view in Lotze's Logic.
BOOK III.
POSITIONAL MANIFOLDS.
CHAPTER I.
Fundamental Propositions.
61. Introductort. (1) In all algebras of the numerical genus (of. § 22)
any element of the algebraic manifold of the first order can be expressed in
the form a^^i + 0,6^ + . . . + a,e,, where ei, e^,... e^ are v elements of this manifold
and Oi, a,, . . . a, are numbers, where number here means a quantity of ordinary
algebra, real or imaginary. It will be convenient in future invariably to use
ordinary Roman or italic letters to represent the symbols following the laws
of the special algebra considered : thus also each group of such letters is a
symbol following the laws of the special algebra. Such letters or such group
of letters may be called extraordinaries* to indicate that in their mutual
relations they do not follow the laws of ordinary algebra. Greek letters will
be strictly confined to representing numbers, and will in their mutual
relations therefore follow all the laws of ordinaiy algebra.
(2) The properties of a positional manifold will be easily identified with
the descriptive properties of Space of any number of dimensions, to the
exclusion of all metrical properties. It will be convenient therefore, without
effecting any formal identification, to use spatial language in investigating
the properties of positional manifolds.
A positional manifold will be seen to be a quantitively defined manifold,
and therefore also a complex serial manifold (cf. § 11).
(3) The fundamental properties which must belong, in some form or other,
to any positional manifold must now be discussed. The investigation of §§ 62
63 will be conducted according to the same principles as that of §§ 14 — 18,
which will be presupposed throughout. The present investigation is an
amplification of those articles, stress beiug laid on the special properties of
algebraic manifolds of the numerical genus.
62. Intensity. (1) Each thing denoted by an extraordinary, repre-
senting an element of a positional manifold, involves a quantity special to it,
to be called its intensity. The special characteristic of intensity is that in
general the thing is absent when the intensity is zero, and is never absent
* This name was used by Cayley.
120 FUNDAMENTAL PROPOSITIONS. [CHAP. I.
unless the intensity is zero. There is, however, an exceptional case discussed
in Chapter iv. of this book.
(2) Two things alike in all respects, except that they possess intensities
of diflFerent magnitudes, will be called things of the same kind. They repre-
sent the same element of the positional manifold, the intensity being in feust
a secondary property of the elements of the manifold (cf. § 9).
(3) Let any arbitrary intensity of a thing representing a certain element
be chosen as the unit intensity, then the numerical measure of the intensity
of another thing representing the same element is the ratio of its intensity
to the unit intensity. Let the letter e denote the thing at unit intensity,
then a thing of the same kind at intensity a, where a is some number, will
be denoted by ae or by ea, which will be treated as equivalent symbols.
(4) Let the intensity of a thing which is absent be denoted by 0. Then
by the definition of intensity,
(5) Further, two things representing the same element at intensities a
and /S are to be conceived as capable of a synthesis so as to form one thing
representing the same element at intensity a-vfi. This synthesis is un-
ambiguous and unique, and such as can be symbolized by the laws of
addition. Hence
a« + i86 = (a + i8)e = (i8 + a)e=i86+ 05.
The equation
involves the formal distributive law of multiplication (cf. § 19). Accordingly
in the symbol 06, we may conceive a and e, as multiplied together.
(6) Conversely a thing of intensity a + /S is to be conceived as analysable
into the two things representing the same element at intensities a and /8.
Then it is to be supposed that one of the things at intensity /S can be
removed, and only the thing at intensity a left. This process can be con-
ceived as and symbolized by subtraction. Its result is unambiguous and
unique. Hence
(a + /8)6 — i86 = a6.
(7) If corresponding to any thing ae there can be conceived another
thing, such that a synthesis of addition of the two annihilates both,
then this second thing may be conceived as representing the same element
as the first but of negative intensity — a [c£ § 89 in limitation of this
statement].
Thus ae + (- ae) = ae - oe= Oe = 0.
Complex intensities of the form a + iyS can also be admitted (i being V^l).
It was explained in § 7 that the logical admissibility of their use was
altogether independent of the power of interpreting them.
62, 63] INTENSITY. 121
(8) Thus finally, if a, /8, 7, S be any numbers, real or complex, and e
an extraordinary, we have
oBe + /3Se — ySe = (aS + /8S — 7S) e = S (a + /8 — 7) c = S (ae + /Sc — ye) ;
also Oe = 0.
All the general laws of addition and subtraction (cf. §§ 14 — 18) can be
easily seen to be compatible with the definitions and explanations given
above.
(9) It must be remembered that other quantities may be involved in a
thing ae besides its own intensity. But such quantities are to be conceived
as defining the quality, or character, of the thing, in other words, the element
of the manifold which the thing represents ; as for instance its pitch defines
in part the character of a wrench. If any of these quantities alter, the thing
alters and either it ceases to be capable of representation by any multiple of
e, or e can represent more than one element [cf. § 89 (2)].
63. Thinqs representing different elements. (1) Let 61, es...^,.
denote v things representing different elements each at unit intensity. Let
things at any intensities of these kinds be capable of a synthesis giving a
resultant thing ; and let the laws of this synthesis be capable of being sym-
bolized by addition.
Let a be the resultant of a^^, 0,62, ... a^e^ ;
then a = aiei + aj6i + a,e3+ ... +a^e^.
(2) By these principles and by the previous definitions of the present
chapter,
2a = a + a = {a^Si + 0,61 + . . . + a^e^) + (ttjei + . . . + a^e,)
= («! + cii) ^ + (a, + a,) e, + . . . + (a„ + «„) e^
= 2ai6i + 2a,ej 4- . . . + 2a„c^.
Similarly if /8 be any real positive number, integral or firactional,
/8a = /SaiBi + ySojCj 4- . . . + ffoL^e^.
Let this law be extended by definition to the case of negative and complex
numbers.
Hence for all values of /3
y9a = /8(aiei + ajes+ ... ar^i') = /8ai6i 4-/80,6,+ ... + I3a,e^.
Then Oa= Oei 4- Oe, 4- ... 4- 06„ = 0.
(3) The resultant of an addition is a thing possessing a character
(in that it represents a definite element) and intensity of its own. The
character is completely defined (cf. Prop. n. following) by the ratios
Hence the intensities are secondary properties of elements according to the
definition of § 9.
The comparison of the intensities of things representing different elements
122 FUNDAMENTAL PROPOSITIONS. [CHAP. I.
may be possible. The whole question of such comparison will be discussed
later in chapter iv. of this book. But it is only in special developments that
the comparison of intensities assumes any importance: the more general
formulae do not assume any definite law of comparison.
(4) Definition of Independent Units, Let ei, ej-..^^ be defined to be
such that no one of them can be expressed as the sum of the rest at any
intensities. Symbolically this definition states that no one of these letters,
ei, say, can be expressed in the form a^ + 0^9 + ... + a^e^.
Then ei, es ... ^^ ^^^ said to be mutually independent. If 61, es ... 6,, are
all respectively at unit intensity, then they are said to be independent units.
Any one of them is said to be independent of the rest.
64. Fundamental Propositions. (1) A group of propositions* can
now be proved; they will be numbered because of their importance and
fundamental character.
Prop. I. If ^1,02... e,, be v independent extraordinaries, then the equation,
ai^i + Ojej + ... + a^„ = 0,
involves the n simultaneous equations, ai = 0, ^ =s 0 ... a^ = 0.
Suppose firstly that all the coefficients are zero except one, a, say, then
a^ei = 0. And by definition this involves ai = 0.
Again assume that a number of the coefficients, including Ui, are not
zero. Then we can write
e, = ^^&.-^e - -^^
But this is contrary to the supposition that ^i, e, . . . 6,, are independent.
Hence finally all the coefficients must separately vanish.
Prop. II. If the two sums ai^i + aj6j + . . . + a„e„ and A^i + /836a + . . . + 0^,
are multiples of the same extraordinary, where 01,6,...^,, are independent
extraordinaries, then aj//3i = a,/ A = . . . = ajff^.
For by hypothesis Aei + i8a6a+ ... +/3^^ = 7(ai«i + ajej+ ... +a^,.).
Hence (J3i-yai)ei-{'(/3i-ya^)e^+ ,..+(/3^'-ya^)e^ = 0.
Therefore by Proposition I, A - 7ai = 0, A - 702 = 0 . . . i8„ - 7a^ = 0.
Hence A/tti = iS^o, = . . . = /8„/a„ = 7.
It follows [c£ § 62 (2)], as has been explained in § 63 (3), that the ratios
of the coefficients of a sum define the character of the resultant, that is to
say, the element represented by the resultant. Only it must be remembered
that the extraordinaries have to be independent.
(2) These propositions make a few definitions and recapitulations desirable.
If two terms a and 6 both represent the same element, but at diflFerent
intensities, then a and 6 will be said to be congruent to each other. The
* Gf. Graasmann, AtudehnungsUhre of 1862 ; also De Morgan, Transactions of the Cambridge
Philosopkieal Society, 1844.
64] FUNDAMENTAL PROPOSITIONS. 123
fact that the extraordiDaries a and h are congruent will be expressed by
a = 6. This relation implies an equation of the form a = X6, where X is some
number. The sjrmbol = will also be used to imply that an equation, concerned
solely with the quantities of ordinary algebra, is an identity.
(3) The extraordinary ai«i + a^ + . . . + dve^ will be said to be dependent
on the extraordinaries ^i, e, ... 0„; and the element represented by
will be said to be dependent on the elements represented by 6i, e, ... e^,
An expression of the form tti^i + (Lfi^ + . . . + ct^^ is often written Sac.
(4) Let the v given independent extraordinaries be called the original
defining extraordinaries, or the original defining units, if they are known
to be at unit intensity. They define a positional manifold of i^ — 1 dimensions
(cf. § 11). Any element of the type 2a« belongs to this manifold. This
complete positional manifold, found by giving all values (real or complex) to
ai, a,, ... fty, will be called the complete region. Any p of these v defining
extraordinaries define a positional manifold of p — 1 dimensions. It is
incident in the complete region, and will be called a subregion of the complete
region.
A region or subregion defined by 6,, 6j... Cp will be called the region or
subregion (6i,c...£?p).
(5) As far as has been shown up to the present, the v defining units
represent elements which appear to have a certain special function and
preeminence in the complete region. It will be proved in the succeeding
propositions that this is not really the case, but that any two elements are
on an equality like two points in space.
(6) If letters a, 6, c ... denoting elements of the region be not mutually
independent, then at least one equation of the form,
aa + i86 + 7C+ ... = 0,
exists between them, where a, /8, 7 ... are not all zero.
Let such equations be called the addition relations between the mutually
dependent letters.
(7) Prop. III. An unlimited number of groups of v independent extra-
ordinaries can be found in a region of 1/ - 1 dimensions.
Let the region be (ci, e^ ... e^).
It is possible in an unlimited number of ways to find v* numbers, real or
complex, tti, 03 ... a^, /3i, ^Ss ... /Sy ... Ki, ^ ... k^, such that the determinant
Cti, CC3 . . . (X|f
Pit Pi •*• P»
fvi • 1.0 ... K.
'1>
is not zero.
124 FUNDAMENTAL PROPOSITIONS. [CHAP. I.
Let a = ai6i + 0,6^ + ... + a,^v,
Now let f , i; . . . ;^ be numbers such that (if possible)
fa + 176 + ... + xA? = 0.
Then substituting for a, 6 ... A;, we find
Hence by Proposition I, the v coefficients are separately zero.
But since the determinant written above is not zero, these v equations
involve ^ = 0, i7 = 0...;^ = 0. Hence the v letters a, 6 ... A; are mutually
independent.
Prop. IV. No group containing more than v independent letters can be
found in a region of i/ — 1 dimensions.
Let the region be defined by Ci, eg...^,,, and let Oi, a^.a^ be v in-
dependent letters in the region. Then by solving for ei, e^ ... 6^ in terms of
Oj, Os ... a,, we can write
ei = OiiOi + auOg + ... + cri/t„,
Bp = CL^di + *»A + ... + CtyrOy.
Now any other letter b in the region is of the form
hence substituting for 6i, e, ... ^,. in terms of Oi, a, ... a,.,
6 = YiOi + YaOj + ... + 7/t^.
Thus 6 cannot be independent of Oi, a, ... a^.
«
Prop. V. If Oi, a,... ap be p independent extraordinaries in a region of
i; — 1 dimensions, where v is greater than p, then another extraordinary can be
found in an infinite number of ways which is independent of the p independent
extraordinaries.
Let the complete region be defined by 6i, eg, ... e^. Then the expressions
for Oi, Os ... dp in terms of the units must involve at least p of the defining
extraordinaries with non-vanishing coefficients in such a way that they
cannot be simultaneously eliminated. For if not, then the defining extra-
ordinaries involved define a region containing Oi, a^, ...a^ and of less than
p — 1 dimensions. But since Oi, a,, ... Op are independent, by Prop. IV. this
is impossible.
\
64, 65] FUNDAMENTAL PROPOSITIONS. 125
Let the extraordinaries ei, 6,, ... ^p at least be involved in the expressions
for Oi, a,, ... dp. Then by solving for 6i, e, ... Cp, we have
$1 = auOi + auO, + . . . + aipOp + aj ^p+iCp+j + . . . + fli^c^,
Op = ttpiCti 4- ApsG^s + • • • + ^ppdfi + ttp.p+i^p+i + . . . + OLp^ey.
Let 6 be any other letter, defined by
Substituting for ^, 6, ... 6p,
6 = ^Oi + ... + lypCip + ?i+i«p+i + ... + ?i.^F ;
where any one of the f 's, say 5'^, is of the form
Thus there are v undetermined numbers, fi, fa ... fr, and v—p coefficients of
Cp+i, 6(p+a ... e„ (viz. tp+i> ••• W' Hence it is possible in an infinite number of
ways to determine the numbers so that all these coefficients do not simul-
taneously vanish. And in such a case h is independent of the group ai . . . a^^.
Corollary. By continually adding another independent letter to a
group of independent letters, it is obvious that any group of independent
letters can be completed so as to contain a number of letters one more than
the number of dimensions of the region.
(8) By the aid of these propositions it can be seen that any group of v
independent extraordinaries can be taken as defining the complete region.
The original units have only the advantage that their unit intensities are
known (c£ chapter iv. following).
Any group of v independent elements which are being used to define a
complete region will be called coordinate elements of the region, or more
shortly coordinates of the region.
66. SUBREQIONS. (1) The definition of a subregion in § 64 (4) can be
extended. A region defined by any p independent letters lying in a region of
y — 1 dimensions, where p is less than i/, is called a subregion of the original
region. The original region is the complete region, and the subregion is
incident in the complete region.
(2) A region of no dimensions consists of a single element. It is
analogous to a point in space.
A subregion of one dimension, defined by a, 6, is in its real part the
collection of elements found from fa + 17&, where f/17 is given all real values.
Hence it contains a singly infinite number of elements, which will be called
real elements when a and h are considered to be real elements. It is analo-
gous to a straight line. And like a straight line it is given an imaginary
extension by the inclusion of all elements found by giving f /17 all imaginary
valuea
A subregion of two dimensions, defined by a, 6, c, is the collection of
126 FUNDAMENTAL PROPOSITIONS. [CHAP. I.
elements found fix)m f a + 176 + fc, where f /f , ly/iT are given all values. Hence
its real part contains a doubly infinite number of elements. It is analogous
to a plane of ordinary space.
Similarly a subregion of three dimensions contains in its real part a trebly
infinite number of elements, and is analogous to space of three dimensions ;
and so on.
(3) A subregion is called a co-ordinate region when, being itself of /o — 1
dimensions, p of the co-ordinate (or reference) elements of the complete region
have been taken in it.
For example, if e,, e^.^.e^ be the co-ordinate elements, then the region
defined by 61, 0, ... ^p is a co-ordinate region of p — 1 dimensions.
(4) The complete region being of 1/ — 1 dimensions, to every co-ordinate
subregion of p — 1 dimensions there corresponds another co-ordinate subregion
S ^ . of (1/ — /o — ^) dimensions, so that the two do not overlap, and the co-ordinate
"^ elements of the two together define the complete region. Such co-ordinate
regions will be called supplementary, and one will be said to be supplementary
to the other.
(5) If two co-ordinate subregions of /o — 1 and o- — 1 dimensions do not
overlap, then there is in the complete region a remaining co-ordinate sub-
region of (1/ — /o — <7 — 1) dimensions belonging to neither; where of course
/o + <7 is less than v.
If the co-ordinate regions do overlap and have a common subregion of t — 1
dimensions, then the remaining co-ordinate subregion isof(i/ + T — p — cr — 1)
dimensions ; also the common subregion of r - 1 dimensions must be a
co-ordinate region.
If /o + ^r is greater than r, then the subregions must overlap and have in
common a subregion of at least {p + a — v — 1) dimensions.
(6) Let two regions of /o — 1 and <7 — 1 dimensions overlap and have in
common a subregion of r — 1 dimensions. Let the subregion be defined by
the terms Oi, a, ... Or- Then the region of p — 1 dimensions can be defined by
the p terms
dl, (Zs . . . dry Ot+1 ... Op J
and the region of o- ~ 1 dimensions by the a terms
Or\ , Cb^ ... iMtf , Cf-\-\ ... C^ I
where the t«rms ft^+i ••• 6p are independent of the terms c^+i ... c^. For if the
6*8 and the c's do not together form a group of independent letters then
another common letter 0^+1 can be found independent of the other a's and can
be added to them to define a common subregion of r dimensions.
The region defined by the letters
Oriy CL^ ..• d^f O^-f-i ••* Op, Cp^i ... C^
is called the containing region of the two overlapping regions ; and is of
(p + a — T — 1) dimensions.
65] SUBREOIONS. 127
It is the region of fewest dimensions which contains both regions as sub-
regions, whether the regions do or do not overlap.
(7) Every complete region and every subregion can be conceived of as a
continuous whole. For any element of a subregion can be represented by
a? = f lOi + fsO, + . . . + ^pa^ ;
and by a gradual modification of the values of the coefficients x can be
gradually altered so as to represent any element of the region. Hence x can
be conceived as representing a gradually altering element which successively
coincides with all the elements of the region. The region can always there-
fore be conceived as continuous.
(8) Also the subregions must not be conceived as bounding each other.
Each subregion has no limits, and may be called therefore unlimited. For
any region is an aggregation of elements, and no one of these elements is
more at the boundary or more in the midst of the region than any other element.
Overlapping regions are not in any sense bounded by their common subregion.
For any subregion of a region may be common to another region also.
Begions, therefore, are like unlimited lines or surfaces, either stretching in
all directions to infinity or returning into themselves so as to be closed ; two
infinite planes cutting each other in a line are not bounded by this line, which
is a subregion common to both.
(9) Consider the one-dimensional subregion defined by the two elements
Oi and a,. Any extraordinary w which represents an element belonging to
the subregion is of the form fiOi-hfta,. As ^^/^i takes all real positive
values fix)m 0 to +00 , a may be conceived as representing a variable
element travelling through a continuous series of elements arranged in
order, starting from Oi and ending at a,. Again, as ft/fi takes all real
negative values from —00 to 0, ^ may be conceived as travelling through
another continuous series of elements aiTanged in order, starting from a, and
ending in Oi.
It is for the purposes of this book the simplest and most convenient sup-
position to conceive a one-dimensional subregion defined by two elements as
formed by a continuous series of elements arranged in order, and such that
by starting from any one element Oi and proceeding through the continuous
series in order a variable element finally returns to Ui.
This supposition might be replaced* in investigations in which the object
was to illustrate the Theory of Functions by another one. The element
X starts from Oi and passes through the series of elements given by fs/^i
positive and varying from 0 to x , and thus reaches a, ; then as f^/f 1 varies
from — 00 to 0, a; passes through another series of elements and finally reaches
an element which in our symbolism is Oj. But Oi, ds Ihus arrived at, may be
conceived to be a different element firom the element Oi from which x
started.
* Cf. Klein, Nieht-Euhliditche Qeometrie, Vorlesungen, 1889—1890.
128 FUNDAMENTAL PROPOSITIONS. [CHAP. I.
This conception has no natural symbolism in the investigations of this
work, and therefore will not be adopted ; but in other modes of investigation
it is imperative that it be kept in view. Call this second Oi the Oi of the
first arrival, and denote it by lOi. Similarly we might find an a, of the
second arrival and denote it by jCii, and so on. Finally the analysis might
suggest the identification of the Oi of the /oth arrival with the original Oj.
Thus Oi = pOi. It is sufficient in this treatise simply to have noticed these
possibilities.
66. Loci. (1) A locus is a more general conception than a subregion ;
it is an aggregation of a number (in general infinite) of elements deter-
mined according to some law. Thus if x denote the element f jCi + ^^ 4- . . . f ^r,
in the region defined by e^, €^..,6^, then the equation <^ (fi, fj ... fr) = 0,
where ^ is a homogeneous function, limits the arbitrary nature of the ratios
fi : fj ... : f,^ The corresponding values of x form therefore a special aggre-
gation of the elements out of the whole number in the region. But these
elements, except in the special case of flat loci (cf. subsection (6) of this
section), do not form themselves a subregion, according to the definition of
a subregion given in this book; they are parts of many subregions.
(2) A locus may be defined by more than one equation: thus the
equations ^(fi ... fi.) = 0, ^(fi... f,.) = 0, ..., ^p(fi... fr) = 0, where the
left-hand sides are all homogeneous, define a locus when treated as simul-
taneous. If there be r — 1 independent equations, they determine a finite
definite number of elements; and more than v-1 equations cannot in
general be simultemeously satisfied.
(3) A locus defined by p simultaneous equations will be said to be of
v — p — l dimensions when the case is excluded in which the satisfaction of
some of the equations secures that of others of the equations. In a region
of v — I dimensions there cannot be a locus of more than v — 2 dimensions,
and a locus containing an infinite number of elements must be of at least
one dimension. Hence such a locus cannot be defined by more than r — 2
equations. In a locus of one dimension the number of elements is singly
infinite : in a locus of two dimensions it is doubly infinite, and so on.
(4) Let p + a equations define a locus of v — p — a — l dimensions.
These equations may be split into two groups of p equations and of c
equations respectively. The group of p equations defines a locus ot v — p — 1
dimensions, and the group of <r equations defines a locus of i/ — o- — 1 dimen-
sions. The original locus is contained in both these loci. Hence the locus
of I/ — p — <7 — 1 dimensions may be conceived as formed by the intersection
of two loci of v - /o — 1 dimensions and of i/ — <r — 1 dimensions respectively.
Similarly these intersecting loci can be split up into the intersections of
other loci of higher dimensions. So finally the locus of v — p — l dimensions
66] LOCI. 129
may be conceived as the intersection of p loci of i/ - 2 dimensions ; each of
these loci being given by one of the simultaneous equations.
(5) The locus corresponding to an equation of the first degree, namely,
aif 1 + Ojf , + . . . + a^f ^ = 0,
is also a subregion of i; — 2 dimensions, as well as being a locus of the same
number of dimensions.
For i£ Xi,(c^...x,he V elements in the locus, given by
^ = f 11^ + Sufii + . . . giwBp ,
then the v equations of the type,
a,fpi + a^f^ + ... + a,^p^ = 0,
involve the vanishing of the determinant
in fu ••• %iv
in (aa ••• vav
ivi iv% • • • > rr
Hence an addition relation of the form,
\pi^ + XjiTj + . . . + X,^,, = 0,
exists between the elements. The v elements are therefore not independent,
But I' — 1 independent points can be determined. Again, since the equation
of the locus is linear, if Xi and a^ be two elements in it, then X^ + /lut, also
lies in the locus. Hence the whole region of the i/ — 1 independent elements
is contained in the locus, and vice versa ; the region and locus coinciding.
A locus defined by p simultaneous independent linear equations can in the
same way be proved to contain groups oi v — p independent elements. It
therefore in like manner can be proved to be a subregion of v — p — 1
dimensions. Let such a locus be called ' flat.' Then a flat locus is a sub*
region.
(6) A locus defined by p + o- equations, of which p are linear, can
therefore be treated as a locus of v — p — a — 1 dimensions in a region of
j^ — p — 1 dimensions.
(7) There is a great distinction between the region defined by the com-
bined elements which define subregions and the flat locus determined by
the simultaneous satisfaction of the equations of other flat loci. Consider
for instance, in a complete region of two dimensions, the two subregions
defined by ^, e^ and e,, e^ respectively. The region defined by the four
elements ej, e,, 6^, 64 includes not only the elements of the subregion ei^,
of the form feei + faet, and of the subregion 6^4, of the form f>e, 4-^4^4;
but also it includes all elements of the form
fi«i-f-faei+fae8 + f4«4,
w. 9
130 FUNDAMENTAL PROPOSITIONS. [CHAP. I.
which includes elements not lying in the subregions. But the equations of
the loci taken together indicate a locus which is the region (in this case a
single element) common to the two regions eie^y e^^.
67. Surface Loci and Curve Loci. (1) Let a locus which is of one
dimension less than a region or subregion containing it be called a surface
locus in this region. Let this region, which contains the surface locus and is
of one dimension more than the surface locus, be called the containing region.
In other words [cf. § 66 (6)] a surface locus can be defined by p + 1 equations,
of which only one at most is non-linear and p (defining the containing region)
are linear. A surface- locus defined by an equation of the fith degree will
be called a surface locus of the fxth. degree. For example, in this nomen-
clature we may say that a surface locus of the first degree is flat.
(2) In reference to a complete region of i; — 1 dimensions a flat surface
locus of i^ — 2 dimensions will be called a plane. A flat locus of i/— 3 dimen-
sions will be called a subplane in a complete region of v — 1 dimensions.
Subplanes are planes of planes.
(3) A subregion is either contained in any surface locus of the complete
region or intersects it in a locus which is another surface locus contained in
the subregion.
Let the complete region be of v — 1 dimensions and the subregion of
I/ — p — 1 dimensions. Then the subregion is a flat locus defined by p
equations. The intersection of the subregion and the surface locus is there-
fore defined by p + l equations and is therefore of i^ — p — 2 dimensions.
Also it lies in the subregion which is of p — p — 1 dimensions. Hence it is
a surface locus; unless the satisfaction of the p equations of the flat locus
also secures the satisfaction of the equation of the surface locus. In this
case the subregion is contained in the ori^^al surface locus.
(4) A locus of one dimension either intersects a surface locus in a
definite number of elements or lies completely in the surface locus. For if
the complete region be of i/ — 1 dimensions, the locus of one dimension is
determined by i/ — 2 equations ; and these together with the equation of the
surface locus give i^ — 1 equations which, if independent, determine a definite
number of elements. If the equation of the surface locus does not form an
additional independent equation, it must be satisfied when the equations of
the one dimensional locus are satisfied ; that is to say the one dimensional
locus lies in the surface locus.
A locus of one dimension will be called a curvilinear locus. A flat curvi-
linear locus is a region of one dimension, and will be called a straight line.
(5) A locus of T — 1 dimensions which cannot be contained in a region
of T dimensions (i.e. which is not a surface locus) will be called a curve locus.
Thus if the locus be determined hy p + a equations, of which p are linear
and <r non-linear, then o- > 1 ; and the locus is ofi/ — p — <r — 1 dimensions.
67] SURFACE LOCI AND CURVE LOCI. 131
(6) A curve locus formed by the intersection of p surface loci, each
in the same containing region, and such that it cannot be contained in
a region of fewer dimensions than this common containing region, will be
called a curve locus of the (p — l)th order of tortuosity contained in this
region. Thus the order of tortuosity of a surface locus is zero.
Each of the surface loci, which form the curve locus by their intersection,
will be called a containing locus.
(7) In general the intersection of a curve locus of any order of tortuosity
with a subregion is another curve locus contained in that subregion and of
the same order of tortuosity. For the curve locus may be conceived as
defined by p equations of the first degree which define the containing region
of V — p — 1 dimensions; and by a equations
each of a degree higher than the first degree, which define the tortuosity.
Now the subregion which intersects the curve locus is defined by t equations
of the first degree in addition to the equations defining the containing
region. These p + r equations of the first degree and the a equations
^1 = 0, ... ^a = 0, in general define a curve locus of the (a — l)th order of tor-
tuosity in the subregion.
(8) A plane curve in geometry is a sur£Bu;e locus ; a plane curve of the
first order of tortuosity consists of a finite number of points.
In three dimensions an ordinary surface is a surface locus, an ordinary
tortuous curve is a curve locus of the first order of tortuosity ; and a finite
number of points not in one plane form a curve locus of the second order
of tortuosity.
There are of course exceptional cases in relation to the tortuosity of
curve loci when the equations are not all independent. It is not necessary
now to enter into them.
Note. The analytical part of this chapter follows closely the methods of Graes-
mann's Attsdehnungdekre von 1862, chapter i. §§ 1 — 36. This theory of Grassmann is a
generalization of Mobius' Der Barycentrischer Calcul (1827), in which the addition of points
is defined and considered. Hamilton's Quaternions also involve the same theory of the
addition of extraordinaries (the number of independent extraordinaries being however
limited to four). This theory is considered in his 'Lectures on Quaternions' (1853),
Lecture I, and in his * Elements of Quaternions/ Part I. The idea in Hamilton's works
was a generalization of the composition of velocities according to the parallelogram law.
Hamilton's first paper on Quaternions was published in the Philosophical Magazine (1844) ;
De Morgan in his last paper, ' On the Foundation of Algebra' (loc. cit 1844) writes of it,
< To this paper I am indebted for the idea of inventing a distinct system of unit-symbols,
and investigating or assigning relations which define their mode of action on each other.'
Some simple ideas which arise in the study of Descriptive Geometry of any number
of dimensions have been discussed in §§ 66, 67 as far as they will be wanted in this
treatise. On this subject Cayle/s * Memoir on Abstract Geometry,' Phil. Trans. YoL clx.
1870 (and Collected Mathematioal Papers^ Vol. vi. No. 413), should be studied. It enters
into the subject with a complete generality of treatment which is not necessary here.
9—2
CHAPTER 11.
Straight Lines and Planes.
68. Introductory. The theorems of Projective Geometry extended to
any number of dimensions can be deduced as necessary consequences of the
definitions of a positional manifold. Qrassmann's 'Calculus of Extension' (to be
investigated in Book IV.) forms a powerful instrument for such an investigation ;
the propei-ties also can to some extent be deduced by the methods of ordinary
co-ordinate (Jeometry. Only such theorems will be now investigated which
are either useful subsequently in this treatise or exemplify in their proof the
method of the addition of extraordinaries.
69. Anharmonic Ratio. (1) Any point p on the straight line aa
can be written in the form fa + f V, where the position of p is defined by the
ratio f/^. If pi be another point, fia + f I'a', on the same line, then the
ratio f fZ/ff 1 is called the anharmonic ratio of the range {axi\ pp^). It is to
be carefully noticed that the anharmonic ratio of a range of four collinear
elements is here defined apart from thie introduction of any idea of distance.
It is also independent of the intensities at which a and a' happen to repre-
sent their elementa For it is obviously unaltered if a, a' are replaced by
aa, a'a', a and a' being any arbitrary quantities.
(2) If the anharmonic ratio of (cwt', ppi) be — 1, the range is said to be
harmonic ; and p and pi can then be written respectively in the forms
fa + f V and fa - f a .
(3) Let pi, Pj, jpa, p^ be any four points, fia + f/a', etc. Then
where (f if a') stands for the determinant f if g' - f jf /.
Similarly a' = (fi;>. - f,pi)/(fif /).
Hence p, = f 3a + f ,V = {(f ,f /) p, - (faf /) p.}/(f if 0 ;
and p, = Kf ,f ;) p, - (f ,f 0 pa}/(f if /).
Hence the anharmonic ratio of the range {pip^, PzPd is
68 — 71] HOMOGRAPHIC RANGES. 133
70. HoMOORAPHic Bakoes. (1) Let
(bib^PiPiPi'") and (CiC2gi?j?8--:)
be two ranges of corresponding points such that the anharmonic ratio of the
four points (6i6a, l>pPp+i), and that of the corresponding points (CiCj, Jpjp+i)
are equal, where p is any one of the suffixes 1, 2, 3, etc.
(2) It can now be proved that the anharmonic ratio of any four points
(puPni PpPv) of the first range is equal to that of the corresponding points
(qki/i, Jpffa) of the second range.
For let JPp = fp&i + fp't2, 9'p = ^pCi + V^.
Then by definition fpf 'p+Vfp+ilp' = ^p^W^p+i V-
Now replace p in turn by p + 1, p4-2, ... cr — 1, and multiply together
corresponding sides of the equations, so obtained. Finally we deduce
ip^<!l^<^P = VpV</IVcVp ;
and hence by subtracting 1 from both sides,
(fpf /)/f erf; = (V,Vo)/VaV;.
It follows that the anharmonic ratio of any four points {pkPii. PpPo) of the
first range is equal to that of the four corresponding points (?\y^, q^q^) of the
second range.
Such con*esponding ranges are called homographic.
71. Linear Transformations. (1) Let e^ and 6, be any two points, and let
be three given points and any fourth point on one range of points ; also let
be the corresponding points on a second range homographic to the first
range.
Then (a,a/) (K)/(«3«.') (t«iO = (A/S/) (i7A')/(AA') (i/A').
Therefore f/f' and 17/17' are connected by a relation of the form
^f'7 + /^^7' + /*T^ + ^'fV = 0 (A),
where \ fi, p!, \* are constants depending on the arbitrarily chosen points
Oi, Cl^i, <h> bi, 62, 6,.
This equation can also be written in the form
V V
where flu* am On, Om are constants which determine the nature of the trans-^
formation, and p must be chosen so that the point
q = V^ + V^
may have the desired intensity.
184 STRAIGHT LINES AND PLANES. [CHAP. II.
Such transformations as those represented algebraically by equations (A)
or (A^) are called linear transformations. Only real transformations will be
considered, namely those for which the coefficients a^i, a^^, a^, ^^ of equation
(A') are real.
(2) There are in general two points which correspond to themselves on
the two ranges. For by substituting f , f' for 17, V ^^ ^^^ above equations
and eliminating we find
(aii-p)(a«-p)-aMan = 0 (B),
an equation which determines two values of p real and unequal or real and
equal or imaginary; and each value of p determines f : f' and 1; : rj' uniquely.
(3) Let pi and p, be the two roots of this quadratic, and first let them
be assumed to be unequal.
Then by substituting pi in one of the equations (A') a self-corresponding
point di is determined by
f /f = ciuKpi - ttu) = (pi - OnVOji.
Similarly a self-corresponding point d^ is determined by
Let these self-corresponding points be the reference points, so that any
point is determined by fdi + f dj.
Then the equations defining the transformation take the form
^=^=P (C).
V V
By putting v for pi/pi, this equation can be written
vlv' = pil^ (C).
(4) Linear transformations fall into three main classes, according as the
roots are (1) real and unequal, (2) imaginary, (3) equal.
In transformations of the first class the two points di and d^ are real.
Then v is real, and is positive when any point in the first range and its
corresponding point on the second range both lie between di and d^.
(5) In transformations of the second class the two points di and d^ are
imaginary. Then v is complex, and it can be proved that mod. p=l, assuming
that real points are transformed into real points.
For pi and p, are conjugate complexes, and can be written
ae^ and ae~^.
Accordingly p=:pjp^=z^.
Hence mod. i' = 1, and log v = 2iS ; where S is real.
(6) The linear transformations of the first class are called hyperbolic ;
and those of the second class elliptic.
71] LINEAR TRANSFORMATIONS. 135
(7) The third special class of linear transformations exists in the case
when the roots of the quadratic are equal, that is to say when the two points
Oi and a, coincide. Linear transformations of this class are called parabolic.
The condition for this case is that in equation (A), modified by substituting
f and f for rj and rj' respectively, the following relation holds between the
coefficients,
4XV = (/i + /jfy.
Let u be the double self-corresponding point and e any other point, and
let these points replace ei and e^ in subsection (1) above. Then the modified
equation (A), regarded as a quadratic in ^ : ^\ must have two roots infinite :
hence
X = 0, fj,+/jf = 0.
Therefore if a point
be transformed into j (= ^w + v^)*
equation (A) takes the form
/* (^' - rv) + Vf V = 0 ;
that is f/f's: 17/97' + constant (D).
(8) By a linear transformation a series of points Pi» Ptt p» •'• c&n be
determined with the property that the range {p^, j),, j), ...) is homographic
with the range (^2ii>s>P4 •••)•
Firstly, let the linear transformation be elliptic or hyperbolic and let the
co-ordinate points ei, e^ be the pair of self-corresponding points of the two
ranges.
Let Pi = ^ei + ^%.
Then if v be any arbitrarily chosen constant, the points
P2 = v^ei + f e,, ^, = i^^ei + f^a, ...p^^i^^^e, + i\
satisfy the required condition.
(9) Secondly, let the linear transformations be parabolic. Let u be the
double self-coiTesponding point, and let e be another arbitrarily chosen
reference point.
Let Pi = ^ + e, and let 8 be the arbitrarily chosen constant of the trans-
formation.
Then by equation (D) the other points of the range are successively
given by
Pp = (f + p-i8)u + 6.
These results mil be found to be of importance in the discussion in
Book VI., chapter i., on the Cayley-Klein theory of distance.
136 STRAIGHT LINES AND PLANES. [CHAP. II.
72. Elementary Properties. (1) Let the v independent elements
«!, e,, ...e„ define the complete region of v — 1 dimensions. Then the p
elements ei, e^, ...ep{p<v) define a subregion of p — 1 dimensions. Any point
in this region can be written in the form
Thus any point on the straight line defined by ei, Cj can be written
and any point on the subregion of two dimensions (or ordinary geometrical
plane) defined by ei, Ct, e^ can be written
fl^ + 52^8 + S8^8-
(2) Any p-f 1 points x^, fl7j...iCp+i in the subregion ei, ej.-.^p can be
connected by at least one equation of the form
fi^i + faaJ2+ ... + fp+i^^p+i = 0.
Let such an equation be called the addition relation between the de-
pendent points a?i, ajj, ... x^^^.
Thus any three points Xi, x^, x^ on a straight line satisfy an equation of
the form
and similarly for any four points on a two-dimensional subregion.
(3) If 6i, 68 ... e„ be the independent reference units of the complete
region, and any point be written in the form 2^6, then the quantities
fi> fa> ... fi*
are called the co-ordinates of the point.
The locus denoted by
is a plane (i.e. a subregion of r — 2 dimensions).
The intersection of the p planes (jXv)
^11 f 1 + Aji f a + . • . + A,„i f „ = 0,
Aia f 1 + AjB s a "i • • . "^ ^v2 %¥ ^ ^>
^ip fi + ^fa + . . . + y^vpiv — V,
is a subregion of i/ — p - 1 dimensions.
(4) The intersection of i/ — 1 such planes is a single point which can be
written in the form
6l y 62 t . .. 6|»,
Aia> Aj3, ... \^f
■•••■•. ........••f......
^ , K— 1 » Aa, i»— 1 » . . . A„^ ,^1
72] ELEMENTARY PROPERTIES. 137
For instance, in the region eie^e^ of two dimensions, the two straight
lines
intersect in the point
Returning to the general case of i; — 1 planes, it is obvious that their point
of intersection lies on the plane
if the determinant
oLi, 0,, ... ay,
Xi, r— 1, X^, |^-l,••. X„^
1^1
vanishes.
(5) To prove that it is in general impossible in a complete region of
V — 1 dimensions to draw a straight line from any given point to intersect
two non-intersecting subregions of /:> — 1 and o- — 1 dimensions respectively,
where p and a are arbitrarily assigned ; and that, when it is possible, only
one such straight line can be drawn. Since the subregions are non-inter-
secting [cf. § 65 (5)]
p + <r < I/.
Of the reference elements let p be chosen in the subregion of p — 1
dimensions, namely ji, jj, ... jp, and let a be chosen in the subregion of cr — 1
dimensions, namely ki, kif...ka, and v — p — a- must be chosen in neither
region, namely Ci, 6s, ... ^^.p-a.
Let the given point be j> = 2a; + ^/3k + Xye.
Let XQ be any point in the subregion Ji....jp« Then p + XS,^ can be
made to be any point on the line joining p and X^ by properly choosing X.
But if this line intersect the subregion ki...k^,, then for some value of X,
say \>p + Xi2{; depends onki...kc only.
Hence either 71 = 73= ... =#y^p_,= 0, in which case p cannot be any
arbitrarily assigned point ; or p + <r = 1/, and there are no reference points of
the type
61, 6j, ... 0y_p_a'*
Hence we find the condition p + o* = y.
Again p + XjSf j » 2/8A?.
Hence also fi = ai, fi = aj, ... f,, = c^, and Xi = — 1.
Thus the line through p intersecting the two regions intersects them in
S2J and X0k respectively. Accordingly there is only one such line through
any given point p.
188 STRAIGHT LINES AND PLANES. [CHAP. IL
73. Reference-Figures. (1) The figure formed in a region of y — 1
dimensions by constructing the straight lines connecting every pair of v
independent elements is the analogue of the triangle in plane geometry
and of the tetrahedron in space of three dimensions, the sides of the triangle
and edges of the tetrahedron being supposed to be produced indefinitely.-
Let such a figure be called a reference-figure, because its comer points can
be taken as reference points to define the region. Let the straight lines be
called the edges of the figure.
(2) Let «i, ei, ... e„ be the comers of such a figure, and let them also be
taken as reference points. Consider the points in which any plane, such as
fi/tti + fa/Os + •.. + frK = 0,
cuts the edges. The point in which the edge e^e^ is cut is found by putting
f S = 0 = f 4 = . . . = f „.
Hence the point is ai^i — Oae,. Similarly the point in which the edge e^e^
is cut is apCp — a^e^.
(3) Consider the points of the typical form OLpS^ + a^e<r.
then the range
is harmonic. Also any point on the plane defined by the point aiBi^-dge^
and the remaining comers of the reference-figure, namely by 63, 64, ... «r, is
Hence all such planes pass through the point
a^ei + 08^2 + oi^ez + . . . + a^6„.
And conversely the planes through this point and all the comers but two cut
the edge defined by the remaining two comers in the points
The harmonic conjugates of these points with respect to the corresponding
comers are of the typical form ap«p — a^e^, and these points are coplanar
and lie on
(4) The point %ae may be called the pole of the plane 2f/a = 0 with
respect to the given reference-figure, and the plane may be called the polar of
the point.
These properties are easily seen to be generalizations of the fiitniliar
properties of triangles (cf. Lachlan, Modem Pure Geometry, § 110).
(5) Let points be assumed one on each edge, of the typical form
It is required to find the condition that they should be coplanar.
Consider the 1/ — 1 edges joining the corner e^ to the remaining comers.
There are V'-\ assumed points of the typical form
73, 74] REFERENCE-FIGURES. 139
on such edges^ and these points define a plane. It remains, therefore, to
determine the condition that any other point of the form
(where neither X nor p is unity) lies on this plane. It must be possible to
choose fj, fg, ... f„ and 17 so as to fulfil the condition
This requires that the coefficients of ^, «,,...«„ should be separately zero.
Hence if a be not equal to X or /Lt, we find f<r=0; and also the three
equations
fAaiA + |^ai^ = 0, fAaM + ^aA^ = 0, f^a^i + ^a^x = 0.
Hence a^ aj^a^i = — a^ Oj^a^ .
But this is the condition that three points on the edges joining 61,^x1^^
should be coUinear. Hence since 61, 6^, e^ are any three of the comers of the
given reference-figure, the necessary and sufficient condition is that the
assumed points lying on the edges which join any three comers in pairs
should be coUinear.
(6) It follows from (3) of this section that the condition for the con-
currence of the planes joining each assumed point of the form
with the comers, not lying on the edge on which the point itself lies, is for
the three points on the edges joining in pairs eie^efi
ctiKCtki^c^m = otxiflifiafiA.
Hence if any three edges be taken forming a triangle with the comers as
vertices, the three lines joining each assumed point with the opposite vertex
are concurrent.
74. Perspective. (1) The perspective properties of triangles can be
generalized for reference-figures in regions of 1; — 1 dimensions *.
Let eiBi ... e^ and e^e^ ... ej be two reference-figures, and let the v lines
be concurrent and meet in the point g. Then it is required to show that the
corresponding edges are concurrent in points which are coplanar.
Since ^ is in 61^/, ea^'> ••• ^f^/, it follows that
Aq^i "T" A.J Bi ^— Aig^s I A(3 &^ ^ . . . *^ A/yCy *7" A»if By ^ g.
Hence X161 — XaCj = XjV — \Wf with similar equations.
But Xi^i-Xjea is on the edge ^i^a, and XjV — X^V is on the edge «iV-
But these are the same point. Hence the edges BiB^ and CiV are concurrent
in this point.
* The theorems of snbseotions (1) to (4) of this article, proved otherwise, were first giyen by
Veronese, cf . " Behandlang der projectivisohen Verhaltnisse der Raume von verschiedenen Dimen-
sionen durch das Princip des Projioirens and Schneidens," Math. Annalen, Bd. 19 (18S2).
140 STRAIGHT LINES AND PLANKS. [CHAP. IL
But taking eie^... e^BB reference points, it has been proved that the points
of the typical form Xi6i — Xs^b ^^ coplanar and lie on the plane, 2^/X = 0.
Hence the theorem is proved.
(2) Conversely, if the corresponding edges, such as e^e, and eiV, intersect
in coplanar points, then the lines 61^1', e^e^^ ... e^e^ are concurrent.
For let two edges such as eie^, ^eiy intersect in the point du. Let the
plane, on which the points such as du lie, have for its equation referred to
fi/Xi + f A + ... + %,tK = 0,
and referred to ^'e,' . . . «/ let it have for its equation
f i7 V + f . W + . . • + f »>»' = 0.
Then any such point c^u can be written
A»i^ •■" Ag^ or A^ ^\ "■ Aj ^ .
But it has not yet been proved that these alternative forms can be assumed
to be at the same intensity. Now consider any corresponding triangles with
comers such as e,ea^ and e^e^e^. Write
<li2 ^^ Ai^i "- Ag^ = ^u (Ai Ci "~ Aj 6j },
Oss = Aj^a — Xs^ = /Cj8 (Xa 6s — Xj ^ ),
Ctji = Xj^ — Xi^i = #Pji (X3 6^ ~" Xi 61 ).
Hence c?ij + d» + c^ = 0, dii/Kj2 + d2ilKn + dn/K^ = 0.
But if these relations are independent, the three points di^^dnydn must
coincide, which is not true. Hence k^ = /Kss = /c^ ; and by altering the in-
tensity of all the points eiC^ ... e/ in the same ratio, each factor such as k,^
can be made equal to — 1.
Hence ciu = X,6i — XaejssXsV — V^' (A).
Thus Xiei + XiV = ^^ + X8V = ". =X„6„ + X/6/ = (7 (B).
Hence the point g is the point of concurrence of
(3) Let the point g be called the centre of perspective and the plane of
the points, dp^, the axal plane of perspective of the two reference-figures.
The equation of the axal plane referred to 6162 ... 6^ is with the previous
notation 2f/X = 0 : its equation referred to ei'e,' ... e/ is 2^7^' = 0* ^^' 9* ^^^
centre of perspective be expressed in the form
(7 = Xiaiei+XjOje, + ... +Xya,,ey.
Then by eliminating ei, es ... e,, by means of equations (B) above,
Xi'^i^i + Xa Oa^j + . . . + X„ OL^e^ = (ai + Oa + . . . 4 a^ — 1) ^.
Hence g^ though of different intensities, can be expressed in the two forms
SXot^ and SXW.
74] PERSPECTIVE. 141
Since ai, as...a, can be assumed in independence of \i, X,, ... X,, it
follows that, given one reference-figure, it is possible to find another reference-
figure in perspective with it having any assigned centre of perspective and
axal plane of perspective.
(4) Suppose that the corresponding edges of three reference-figures
&l v^ • • • vp f v\ v^ • • • &y y &l V% • • • vy
intersect in coplanar points, so that each triad of corresponding edges is
concurrent; and let g, g\ g" be the three corresponding centres of per-
spective.
Consider the three edges e^e^, e^ej, ep'ej\ Then we may assume that
^p^p "" ^0^0 ^ ^p ^p "^ ^9 ^«r *^ '^p ^p "" '*'«r ^«r ^ ^pv^
and hence that
Hence g-^g +g' — 0.
Hence the three centres of perspective are collinear.
(5) Let there be v reference-figures such that each pair is in perspective,
all pairs having the same centre of perspective g. It is required lo show that
all the axal planes of perspective are concurrent.
Let the reference-figures be
Consider the 1/ — 1 pairs of figures formed by taking the first reference-
figure successively with each of the remainder. Let g be the given centre of
perspective, and let the equation of the axal plane of perspective of the pair
comprising the first and the pth figure be, referred to the first figure,
fi/iXip + fV»^p+...+f./^ = 0 (1),
and referred to the pth figure,
fi/iV + &/Api + -- + fi'/Api = 0 (2).
Hence two typical sets of equations are
iXip6u + iXpi^pi = sX]p6u -r 2Xpi^p9 = ... = ¥^^^v + ¥^pi^pv — fl^ ~ *ipfl^] /«\
From equation (1) the point (p) of concurrence of the y — 1 planes of this
type is, when referred to the first figure, given by
P =
l/iXu, 1/jXui ••• I/kXu,
1/iXiK, l/aXii^, ... l/^Xi,,
But by the first of the set of equations (3),
€u = ^ip5^/iXip iXpx^pi/jXip, ... 6iy = ^ipg/p^p "" w^pn^p^l w^^»
142 STRAIGHT LINES AND PLANES. [CHAP. II.
Hence substituting in the expression for p, and noticing that the co-
eflScient of g vanishes, we obtain
P = i^pi^pi/i^ipj a^pi^pa/a^p> ••• » v^pi^pvl^^ipt
This is the point of concurrence of the 1/ — 1 planes, referred to the pt\\
figure.
Now by eliminating ^u, ^u, ... ei„ from equations (3), we obtain
iA.pl iA.iq.gpi — iXgi iA.ipg<yi ^ aA.pl aA'io-gpa ~ 2^<ri aAip^q^
^ip lAa<r — f^iv lA-ip ''^ip aA-io- ''fio- aA-ip
yA.p1 yA>l4y gpy »A.0.1 yA^p gyy
^ip i»A<ia ''^lo- yAqp
Hence the equation of the axal plane of perspective of the pth and ath
figures is, referred to the pth figure,
XlA-pi iA.pi lAq^/ \2A.pi jA.pl jA-i^/
Vi^Api >Api vKiffJ
Now by § 72 (4) the point p lies on this plane, if the determinant formed
by substituting the coefficients of fi, fa, ... f ^ in this equation for
in the determinant, which is the expression for jj, vanishes. The determinant
so formed can be expressed as the sum of two determinants, one with /C]p as a
factor, the other with ^1^ as a factor. The determinant with K^p as a factor
vanishes because it has two rows of the form
■'■/lAip, ^l^npt ••• A/i^A.ip.
The determinant with /Ci^, as a factor vanishes because it has two rows of
the form
Hence all the axal planes are concurrent in the same point.
The particular case of this theorem for triangles in two dimensions is
well-known.
75. Quadrangles. (1) As a simple example of this type of reasoning,
let us investigate the properties of a quadrangle in a two-dimensional region.
Any four points a, 6, c, d are connected by the addition relation
aa + /3b + yc-\-Sd = 0.
Hence aa-^fib and yc + Sd represent the same point, namely the point of
intersection of the lines ab and cd.
75]
QUADRANGLES.
143
(2) Consider the six lines joining these four points. Let the three pairs
which do not intersect in a, 6, c, d, intersect in e,/ g. Then
e = 7C + Ota = — (/86 + 8d),
/= aa + /86 = — (7c + Sd),
g = ^b+ rye = — {Sd + ad).
Hence /— g = oui — yc]
and f+g==l3b-Sd.
From the form of these expressions it follows that/-5r is the point where
fg intersects ac.
Also it follows that e and f—g are harmonic conjugates with respect to a
and c.
Similarly /-{-g is the point where fg intersects bd; and f-i-g and e are
harmonic conjugates with respect to b and d.
Furthermore f—g and f-^g are harmonic conjugates with respect to
/ and g.
The points g ±e, and e +/, have similar properties.
Thus the harmonic properties of a complete quadrilateral are immediately
obvious.
(3) Again the six points f±g,g±e,e±/lie by threes on four straight
lines. For identically
(/-9) + {g-e) + {e-/)=^0.
(/-5') + (S' + «)-(«+/) = 0,
(f+9)-(9-e)-{e+f)^0.
i/+9)-(g + e) + (e-/)='0.
In the accompanjdng figure h and k stand for f^ g respectively, I and r/i
for e T/ respectively, n and p for g ±e respectively.
CHAPTER III.
QUADRICS.
76. Introductory. (1) Let a surface locus of the second degree be
called a quadric surface. Let a curve locus which can be defined as the inter-
section ot p (p< v) quadiic surfaces be called a quadriquadric curve locus. If
it is impossible to define the locus as the intersection of p — cr quadric sur-
faces and of a plane subregions {a < p), then the quadriquadric curve locus is
said to be the (p — l)th order of tortuosity [cf. § 67 (6)].
(2) Let the v reference elements be ^i, 6j, ... e„, and let any point x be
defined by fi^+ ... + ^veyj which is shortened into Sfc.
Let the quadric form aiifi* + 2aiafifs+ ... be written (aja?)'. Then
(a$^? = 0 is the equation of a quadric surface.
Let the lineo-linear form ttnf i% + a^ (f ji/j + f ji/i) + ... be written (a$a;$y ) ;
where x = 2f e, and y = Xrje.
Tf. Elementary Properties. (1) If the element z be of the form
^ + My> then
(a$^)» = \' («$«:)» + 2V («$a:$y) + /*' (a$y)' ; (A).
If more than two elements of a subregion of one dimension lie on a
quadric, the whole subregion lies on it. This follows evidently from
equation (A).
(2) If a quadric contain one plane subregion of the same dimensions
as itself, it must consist of two plane loci taken together. For if there is one
linear fector of a quadric form (cr$a?)* the remaining &ctoT must be linear.
(3) A subregion of any dimensions either intersects a quadric aurSace in
a quadric surface locus contained in that subregion as its containing region or
itself lies entirely in the quadric. For if the subregion be of /:> — 1 dimensions,
it may be chosen as a co-ordinate region containing the p reference elements
6i, es, ... 6p. Hence any element in the region has the vp co-ordinates
76-78] ELEMENTARY PROPERTIES. 145
respectively zero. Thus the intersection of (flf][a?)* = 0 with the subregion is
found by putting these co-ordinates zero in the quadric equation. Thus
either the equation is left as a quadric equation between the remaining p
co-ordinates ; or the left-hand side vanishes identically.
(4) It follows as a corollary from the two previous subsections that a
subregion, which intersects a quadric in one subregion of dimensions lower by
one than itself, intersects it also in another such subregion ; and that these
two flat loci together form the entire intersection of the subregion mth
the quadric.
78. Poles and Polars. (1) The equation, {oi^x\x') = 0, may be con-
ceived as defining the locus of one of the two elements x or of, when the
other is fixed.
If X be fixed, the locus will be called the polar of x' with respect to the
quadric surface, (a$a?)" = 0. The polar of an element is obviously a plane.
The element x' will be called the pole of the plane (a]la?$^) = 0.
(2) The ordinary theorems respecting poles and polai*s obviously hold.
If a; be on the polar of x\ then a/ lies in the polar of x. For in either case
the condition is (a$^$«') = 0. Two elements for which this condition holds
will be called reciprocally polar with respect to the quadric.
If a pole of lie in its polar {oL\x\af) = 0, then
Hence the element x' lies on the quadric. Thus all elements on the quadric
may be conceived as reciprocally polar to themselves : they may be called
self- polar.
The polars of all elements Ijring in a plane must pass through the polar of
the plane.
(3) By oieans of these theorems on poles and polars with respect to any
assumed quadric a correspondence is established between the elements of a
region of v — 1 dimensions and the subregions of y — 2 dimensions.
Corresponding to an element there is its polar subregion : corresponding
to elements lying in a plane of r — 2 dimensions there are polars all containing
the pole of this plane: corresponding to elements lying in a subregion of
i; — p — 1 dimensions there are polars all containing a common subregion of
p — 1 dimensions.
(4) Again, consider the elements in which the linear subregion through
two reciprocally polar elements x and of intersects the quadric. Let \x + fix
be one of these elements, then from equation (A),
V(a][a?)» + /*«(«][«')" = 0.
The two points of intersection must accordingly be of the form \a? ± iiaf.
It follows that two reciprocally polar elements and the two elements in which
w. 10
146
QUADEICS.
[chap. III.
the straight line contaimng them intersects the quadric together form a
harmonic range.
(5) When the element oi is on the quadric, its polar, viz. (a][a?$aj') = 0,
will be called a tangential polar of the quadric. Let m lie on the polar of
any point od on the quadric and let Xx-^-fjuxf be on the quadric. Then
substituting in the equation of the quadric, the equation to determine X//Lt
becomes X' (a$a?)* = 0.
Now in general {(i^ccf is not zero. Hence X is zero and both roots of the
quadratic are zero. Thus all straight lines drawn through an element x'
on the quadric and lying in the polar of x intersect the quadric in two
coincident elements at x\
(6) Let any plane be represented by the equation
The condition that this plane should be a tangential polar of the quadric
is obviously
ttu, CCi2, •.. dwi Xi
^ia> ^a> ••• ^vt Xa
Xi, Xj, ... X,,> 0
This condition can be written in the form
OuV+2ai,XiXa + ...
Let A stand for the determinant
= 0.
= 0.
^IKI ^> ... ff
¥V
then
^ dA ^^ c2A
Now let the plane be denoted by i, then the condition that this plane
may be a tangential polar of the quadric may be written by analogy
Hence corresponding to the condition, (a][a;)«=0, that the element x
lies on the quadric there is the condition, (Oi$Ly = 0, that the plane Z is a
tangential polar of the quadric.
(7) It will be found on developing the theory of multiplication of
Qrassmann's Calculus of Extension (c£ Book iv. ch. i.) that, analogously to
the notation by which an element can be written fi^, + fge, + ... + f„«r where
79] POLES AND POLARS. 147
ei, 62, .*. e^ are extraordinaries denoting reference elements, the plane L can
be written in the form X^Ei + X^E^ +... + \yEy where E^E^.,, Ey are extra-
ordinaries denoting reference planes. Thus the theory of duality will receive
a full expression later [c£ § 110 (4) and § 123] and need not be pursued now,
except to state the fundamental properties.
(8) The equation, (a$a?)"=0, will be called the point-equation of the
quadric, and the equation, (Clt$X)' = 0, will be called the plane-equation of
the quadric. The two equations will be called reciprocal to each other.
(9) It is possible in general to find sets of v independent elements
reciprocally polar to each other.
For let ei be any point not on the quadric. Its polar plane is of i^ — 2
dimensions and does not contain 6i. The intersection of this plane with the
quadric is another quadric of i/ — 3 dimensions contained in it. Take any
point 6s in this polar plane not on the quadric. Again take any point e^ on
the intersection of the polar planes of e^ and e^ ; then e^ on the intersection
of the polar planes of 6], 6a, 6s; and so on. Thus ultimately v independent
points are found all reciprocally polar to each other.
If such points be taken as reference elements, the equation of the quadric
becomes
If the elements lie on the quadric it will be proved in the next article
that v/2 or {v — l)/2, according as v is even or odd, independent elements can
be found reciprocally polar to each other.
79. Generating Regions. (1) A quadric surface contains within it
an infinite number of flat loci, or subregions, real or imaginary, according to
the nature of the quadric. Let such contained regions be called generating
regions. If the complete region be of 2/Lt or of 2/Lt — 1 dimensions, the
subregions, real or imaginary, contained within any quadric surface will be
proved to be of /Lt— 1 dimensions*.
If 6i be any point on the quadric, it lies on its polar (a][6i$a:) = 0.
Let 6j be another element on the quadric lying in the polar of ij. Then
(a][6i$6a) = 0, and each point lies in the polar of the other. Hence any
element Xjbi + Xa^a lies ^ both polars and on the quadric.
But the polars of hi and 6s intersect in a subregion of i' — 3 dimensions,
where v is put for 2ft + 1 or 2/Lt as the case may be.
Take a third point h^ on the intersection of this subregion with the
quadric. Then the three points ftj, 6,, 6, are reciprocally polar, and any point
of the form \i6i + \j6j + X36, lies in the intersection of the three polars and
on the quadric.
* This theorem is due to Veronese, ef. toe. cit,
10—2
148 QUADRICS. [chap. III.
Proceed in this way till p points 61, &,, ... hp are determined such that
each lies on the polars of all the others and on the quadric, and therefore on
its own polar. But the p polars, if 61, &s, ... &p be independent, intersect in
a region of i; — p — 1 dimensions, which contains the p independent points.
Hence ^^P^P^
Hence the greatest value of p is the greatest integer in ^i/.
If r = 2/Lt, or 2/Lt + 1, then p=^fi; there are therefore fi independent points
and these define a subregion of /k ~ 1 dimensions contained in the quadric.
This proposition is a generalization of the proposition that generating
lines, real or imaginary, can be drawn through every point of a conicoid.
(2) If r be even, then each generating region of a quadric is defined by
^if independent points. Hence by § 72 (5) from any point one straight line
can be drawn intersecting two non-intersecting generating regions.
If the point from which the b'ne be drawn be on the quadric and do
not lie in either of the generating regions, the line meets the quadric in
three points, and therefore lies wholly on the quadric.
Hence from any point on a quadric one line and only one line can be
drawn meeting any two non-intersecting generating regions and thus lying
wholly in the quadric.
(3) If V be odd, then each generating region is defined by J (v — 1) inde-
pendent elements. Hence from § 72 (5) it is in general impossible to draw a
line from any point, on or off the quadric, intersecting two non-intersecting
generating regions.
80. Conjugate Co-ordinates. (1) Let the v co-ordinate elements be
a reciprocally polar set. Let the equation of the quadric be
The elements
are on the quadric. They can be assumed to be any two points on the
quadric, not in the same generating region, since ei can be any point not on
the quadric and e^ any point on the polar of ei.
(2) Firstly, let V = 2/A. The set of /Lt elements,
A*^ + (-A)*e4,...
are all on the quadric and reciprocally polar to each other. Hence they define
a generating region on the quadric.
80] CONJUGATE CO-ORDINATES. 149
Similarly the set,
/8i*«i-(-A)*e.,eta
define another generating region on the quadric. Also the /i elements of the
first set are independent of the /t elements of the second set. Therefore the
two generating regions do not overlap at all.
(3) Let the elements of the first set be named in order ji, J2, ... jV, and
of the second set A^, A:,, ... A;^. Then any element x of the form
2£7 = fiji + f,ij+ ... + f^jV
lies in the generating region jij^... jfi, and any element y of the form %r)k
lies in the generating region kik^ ... A:^.
(4) Again, j'l is reciprocally polar to all the k'B except ki. Hence
\Jif ^1 f^i ••• f^ii)
is a generating region, and
\JifJit ^> ^4> ••• ^fi/
is another^ and so on.
Accordingly given one generating region {Jm ]%>'•• jii) including a given
element, other generating regions including that element can be found which
either overlap the given region in that element only, or in regions of
1, 2 ... /A-2
dimensions respectively. Also regions can be found which do not overlap the
given region at all.
(5) The 2/i elements,
Jif Jif ••• Jfii "'i> "^i> »•• iCfi
can be taken as co-ordinate elements. Let them be called a system of
conjugate co-ordinate elements. The properties of such a system are that
they are all on the quadric, and that any pair of elements, with the exception
of pairs having the same suffix, are reciprocally polar, namely j'l not with Ati,^',
not with k^, and so on.
Let ji and ki, j^ and k^, etc. be called conjugate pairs. It can easily be
seen by the method of subsections (1) and (2) of this section that in any
two non-intersecting generating regions of a quadric a conjugate set of
elements can be found, so that ji, j,, ... jV are in one region, and ki, A?,, ... A;^
in the other.
Let any element be written in the form
Then, from the definitions of the conjugate elements in subsections (2) and
(3) above, the equation of the quadric takes the form
150 QUADRICS. [chap. III.
(6) The polar of the element ji is i/i = 0, that is to say, is the region
defined by the elements
The intersection of the polar of ji with the quadric is
and the 2/i — 1 co-ordinate elements
define its containing region.
This quadric is contained in a region of 2/l6 — 2 dimensions. In such a
region quadrics in general have generating regions of /i — 2 dimensions. But
in this quadric all regions of the type
are generating regions, being of /i — 1 dimensions.
(7) The coefficient of ji in the expression defining an element does not
appear in the equation of the quadric. Hence all one dimensional regions
defined by ji and any point on the quadric lie entirely in the quadric. Such
a surface will be called a conical quadric ; the point with the property of ji
will be called its vertex.
(8) Accordingly the intersection of the polar of any element of a quadric
in a region of 2/x - 1 dimensions with the quadric is a conical quadric of
which the given element is the vertex. Thus in three dimensions, the inter-
section of a tangent plane with a quadric is two straight lines, that is to say
a conical quadric in two dimensions.
(9) Secondly, let i/ = 2/* + 1. Then the system of 2/a conjugate elements
JiyJiyJf-9 ^i> ••• ^fi>
C€m be found by the same process as in the first case ; but do not define the
complete region. In forming ji, ...j», k^, ...A?„ from the elements 6i ... e^+i the
last element ej^+i was left over. This element, which will be called simply e,
leaving out the suffix, is reciprocally polar to all the other elements ji ... i^,
but does not lie on the quadric.
(10) Let any element in the region be denoted by
fi ji + ... + f^jV + Vih + ... + Vii^kf, + ^0.
Then the equation of the quadric becomes
fi«h + ^iVi + ... + f^i?^ +(7 = 0.
(11) The polar of the elemental is given by the equation rji = 0.
The intersection of the polar and the quadric is another quadric
which lies in the region j'i,^a, ...jV' ^n ... K, ^. All these co-ordinate elements
are reciprocally polar and all, except e, lie on the quadric.
81] CONJUGATE C0-0R1>1NATES. l5l
This quadric is contained in a region of 2fi — l dimensions, and its
generating regions passing through ji are of /i — 1 dimensions, not more than
the number of dimensions of the generating regions of any quadric in this
region.
(12) Since the coefficient fi of ji does not appear in the equation of the
quadric, if any point x be on the quadric then the region (ji, x) lies entirely
in the quadric. Hence the quadric is a conical quadric.
So finally we find the general proposition that the intersection of the
polar of an element on a quadric with the quadric is a conical quadric with
its vertex at the element.
(13) The reduction of the equation of a quadric contained in a region of
2/i dimensions to the form,
fii7i+...+f^i7M + C' = 0,
is a generalization of the reduction of the equation of a conic section to the
form, LM + ij" = 0 (c£ Salmon's Gonic Sections).
Applying to space of four dimensions the above proposition on the inter-
section of polars with quadrics, we see that if our flat three-dimensional space
be any intersecting region, it intersects the quadric in some conicoid. But if
the space be the polar of some element of the quadric, it intersects the
quadric in a cone with its vertex ^.t the element on the quadric which is the
pole of the space.
A quadric in five-dimensional space has two-dimensional flat spaces as
generating regions.
(14) The co-ordinates ji ... jV> ^i ••• ^m> ^ of a complete region of 2/i
dimensions, giving the equation of some quadric in the form
fii7i + ...+f^i7^ + ir' = 0,
are such that e is the pole of the region ji ... /c^. Now e may be any point in
the region. Hence the polar of any point (Le. any plane) can be defijied by
two not overlapping generating regions of the quadric, viz. ji ... jV ^-^^
ki ... kfiy which are also generating regions of the quadric formed by the
intersection of the polar with the original quadric. This includes the case
of space of two dimensions.
If, however, the complete region be of 2/* — 1 dimensions, the polar of any
point intersects the quadric in another quadric which only contains generating
regions of ft — 2 dimensions ; any two such regions cannot serve to define the
polar which is of 2/i — 2 dimensions. This includes the case of spax^e of three
dimensions.
81. QuADRiQUADRic CuRVE Loci. (1) Consider the general case of
the curve locus formed by the intersection of the p quadric surfaces,
(a,$^)» = 0, {a^Jixy = 0, . . . {a,lxy = 0.
152 QUADRICS. [chap. III.
Let 61 be any point on the locus. Then the polar planes of &i are
(«i$^$^) = 0, (a,$6i$a;) = 0, . . . (ap$6i$a7) = 0.
The intersection of the p polar planes forms a subregion of 1/ — /j — 1
dimensions ; where the complete region is of i^ — 1 dimensions.
The intersection of this region with the curve locus is another quadri-
quadric curve locus of the same order of tortuosity, namely p — 1 [cf § 67 (7)].
Now find another point 62 in this second quadriquadric curve locus, then
all the p polars of 6, contain ftj. Also the 2p polars of 6162 form by their
intersection a region of v — 2p—l dimensions, and this region intersects
the quadriquadric curve locus in another quadriquadric curve locus of the
same order of tortuosity, namely p — 1.
Also it can easily be seen, as in § 79 (1), that the region (bi, b^) lies entirely
in this last curve locus.
Continuing in this way and taking a points bi, 6,, ... 6«r, successively, each
in the quadriquadric curve locus lying in the region of the intersection of
the polars of all the preceding points, we find a subregion defined by
61, 6s, ... 6^, lying entirely in the original quadriquadric curve locus. Also
it must lie in the region of dimensions 1/ — <rp — 1 formed by the intersection
of the polars. Hence we must have
<r ^ 1/ — <rp ;
that is a-< vjifi + 1).
Now let / (\) denote the greatest integer in the number \. Then we
have proved that it is always possible to proceed as above till
<r = /W(p + l)}.
Hence a quadriquadric curve locus, apart from any special relation
between the intersecting quadric surfaces, of tortuosity p — 1, in a complete
region of j/ — 1 dimensions contains subregions (real or imaginary) defined by
/ {i//(p + 1)} elements,
that is to say, of
/ {(1/ — p — l)/(p + 1)} dimensions*.
(2) Hence the least dimensions of a complete region such that a curve
locus, of order of tortuosity p - 1, apart from special conditions must contain
a region of one dimension is 2p + 1.
For example, space of five dimensions is of the lowest dimensions for
which it is the case that the intersection of two quadric surfaces (a curve
locus of order of tortuosity 1) must contain straight lines.
* This generalization of Veronese's Theorem, cf. § 79 1), has not been stated before, as far
as I am aware.
82] QUADRIQUADRIC CURVE LOCI. 163
Also space of eight dimensions is of the lowest dimensions for which it is
the case that a quadriquadric curve locus, of order of tortuosity 1, must
contain subregions of two dimensions.
82. Closed Quadrics. (1) A quadric will be called a closed quadric if
points not on the quadric exist such that any straight line drawn through one
of them must cut the quadric in real points. Such points will be said to be
within the quadric: other points not on the quadric which do not possess
this property will be said to be without the quadric.
(2) Let a straight line be drawn through any point j) cutting the quadric
(a$<r)« = 0.
in two points yi and y^ real or imaginary ; and let x be any other real point
on this line. Also let
then \/fii cuid Xs//a, are the roots of the equation
V(a$p)» + 2V(«$P$^) + A*n«]lL«^)' = 0 (1).
The roots of this equation are real or imaginary according as
is negative or positive.
(3) Now choose as the co-ordinate elements a reciprocally polar system
with respect to the quadric, and let p be one point of this system.
Let the system be j>, ^i, e*, • . . Cp, and let
« = 6> + f,e, + ... + fr^r.
Then {a^xy takes the form
Hence the roots of equation (1) are real or imaginary according as
is negative or positive.
(4) If all lines through p meet the quadric in imaginary points, then the
expression is positive for all values of f,, f,, ... f^. Hence axj, aa,, ... act,
must be all positive; and therefore a, a,, ... a^ must all be of the same sign.
The equation of the quadric takes the form,
and the quadric is therefore entirely imaginary.
(5) Again, if all lines drawn through p meet the quadric in real points,
then flfla, aa„ ... aa^ must all be negative.
164 QUADRICS. [chap. III.
Hence a,, a,, ...7^ are of one sign and a is of the other. The equation of
the quadric can therefore be written in the form
ic,%'+ic,%^+... + fc,%'-f(^? = 0 (2);
where the co-ordinate point p is within the quadric and the remaining v-1
co-ordinate points can easily be proved to be without the quadric on the polar
of p.
(6) It also follows that the polar of a point inside the quadric does not
intersect the quadric in real points. For the polar of |) is f = 0, and its points
of intersection with the quadric lie on the imaginary quadric of j/ — 2
dimensions given by
tC^ ^2 1 • • • i" Kp ^p ^ "•
It has been proved by Sylvester that if a quadric expression referred to
one set of reciprocal co-ordinate elements has p positive terms and v — p
negative terms, then when referred to any other set of reciprocal elements it
still has p positive terms and v — p negative terms (or vice versa).
Hence if the given quadric of equation (2) be referred to any other
reciprocal set of co-ordinates, it still takes the form of (2) as far as the
signs of its terms are concerned. Thus if the quadric considered be a closed
quadric, one element of a reciprocal set of elements is within the quadric and
the remaining elements are without the quadric.
(7) The polar of a point without a closed quadric necessarily cuts the
quadric and contains points within the quadric. For considering the
quadric of equation (2) of subsection (5), the polar of e, is, f, = 0. Its in-
tersection with (2) is the quadric
Ijdng in the plane ft = 0, that is in the region of p, Cs, ... e^. Now if e, be
first chosen, p may be any point in this region and within this quadric, which
is a real closed quadric.
Hence the polar of any point without a closed quadric necessarily cuts the
quadric in real points and contains points within the quadric.
(8) It may be noted that no real generating regions exist on closed
quadric&
(9) Again, choosing any reference points whatsoever, («$«?)• and (a$y)"
are of the same sign if both x and y be inside the closed surface, or if both
be outside the surfSeuse, but are of opposite signs if one be inside the surfeu^e
and one be outside.
For let x' and y' be two points respectively on the polars of x and y. Then
Xx + XW and py^p^y'
are two points on the lines xx* and yy\
83] CLOSED QUADRICS. 155
The points where these lines cut the surface are given by
V (a$a?)* + V« (a$«7 = 0,
and /*'(a][y)"+/*'"(«$y')»=0.
Firstly, let x and y be both within the surface. Then their polars intersect
in a region of i^ — 3 dimensions without the surface. Let x' and y' both
denote the same point z in this region. Then since the roots of the
quadratics for \/\' and fi//jf are both real, (aj-ar)^ has opposite signs to both
(a$a?)» and (ajy)*.
Hence (a^xy and (a][y)* have the same sign.
(10) Secondly let x and y be both without the quadria Then their
polars both cut the quadric, hence x^ and y' may both be chosen within the
quadric. Hence
(a$^')' and (a$y')"
have both the same sign. Also the straight lines xx and yy' both cut the
quadric in real points, since x' and y^ lie within it. Hence ((x^xy has the
opposite sign to (a^x'y and (a$y)' has the opposite sign to (a$yO'. Accord-
ingly (a^xy and (a$y)' have the same sign.
(11) Thirdly, let x be within the quadric and y without the quadric.
Then any point x' on the polar of x lies without the quadric. Also
(a^xy and {a^x'y
have opposite signs, and (cL^xy and (a$y)' have the same sign because both
lie without the quadric. Hence {a^xy and (ajy)* have opposite signs.
83. Conical Quadric Surfaces. (1) To find the condition that
S = (a^xy = 0, should be a conical quadric.
Let b be the vertex and x any point on the surface. Then Xx-^- fib lies on
the surface for all values of X and /ll Hence (cL$b^x) = 0 ; where x is any
point on the surface. Therefore, if 6 = lifie, there are p equations of the type,
found by putting p equal to 1, 2, ... i/, in turn. It follows that the equation
(ajjib'$w) = 0 holds for all positions of x.
And eliminating the fi% the required condition is found to be.
A =
^iv) ^ht^t ••• ®i
rr
= 0.
156
QUADRIC&
[chap. ill.
(2) Also the vei-tex b is the point
^l 9 ^ I * * * ^W
*U> ^1 ••• ^Jv
^l¥t ^rj ••• ^¥P
(3) Again, consider the quadriquadric curve locus of the first order of
tortuosity, defined by
(a$aj)« = 0, and (a'$a;)« = 0.
Any quadric surface intersecting both surfaces in this curve locus is
(a^xy + X (a 5;a?y = 0.
This surface is a conical quadric if
«M + Xfltu', aj, + Xas|^ ... flta^ + XOar'
Hence in general i/ conical quadric surfaces can be di-awn intei*secting
two quadrics in their common curve locua
(4) Let b be the vertex of one of these conical quadrica Take i/ — 1
independent points in the quadriquadric curve locus, so as to make with b an
independent set of elements. Join 6 by a straight line with any one of
these points; the straight line cuts the quadrics again in another common
point. Hence by the harmonic property proved in § 78 (4) it cuts the two
polars of b with respect to the two quadrics in a common point. Hence these
polars have i/ — 1 independent common points. Hence they are identical.
Hence the equations (a][6$ic) = 0, and (a'$6][aj) = 0, are identical.
(5) Let the reference points ei and e, be the vertices of two such conical
quadrics. Then the equations
(a$6i][a?) = 0, and (a^ej^x) = 0,
are identical : that is
Vfi + ««'fc + a„'f.+... + «,/f,= 0| ^'^
are identical.
Similarly the equations
are identical.
84] CONICAL QUADRIC SURFACES. 157
From equations (1) it follows that
ii'
/ "^ / ^ / ^ • • • ^ ^
«U «U *1» ^ir
(3).
firom equations (2) it follows that
/ — " 7 — /—•••— /.....••••.• ••••• \™/»
(Zu ^ ^ ^
Hence either ai, = 0 = a^ ; or else if p be any element \ei + fie^on the line
eiCi, then the two polars
(a$|)$a?) = 0, and («'$?$«) = 0,
are identical Excluding the second alternative, which is obviously a special
case, it follows that any two of the v vertices lie each on the polar of the other
with respect to either quadric. Thus the v vertices form v independent
elements and can be taken as reference elements.
(6) It follows that in general any two quadrics have one common system
of polar reciprocal elements, and that these elements are the v vertices of the
p conical quadrics which can be drawn through the intersection of the two
given quadrics.
(7) Let this system of polar reciprocal elements be taken as co-ordinate
elements. The equations of the quadrics become
and yu%^ + y„V+ • • • + 7r^'^' = 0.
And the ratios 711/71/, 7«/7«'> etc. are the roots, with their signs changed, given
by the above equation [cf. subsection (4)] of the i/th degree determining X.
(8) One means of making the properties of conical quadrics more evident
is to take the vertex as one of the co-ordinate elements of the complete
region. Let 61 be the vertex of the quadric. Then if a; be any element on
the quadric, by hypothesis dei + a; is on the quadric, 0 being arbitrary. Hence
if x = %^, the element
is also on the quadria It follows that ^1 cannot occur at all in the equation
of the quadric. Accordingly the expression (a$a;)' reduces to
84 Reciprocal Equations and Conical Quadrics. (1) When the
quadric (a][d;)'sO is conical, the reciprocal equation of the quadric, namely,
(Oi^Ly^ 0, has peculiar properties.
It has been proved that if b be the vertex, then
whatever element x may be. Hence the polar of any element d? passes
through the vertex 6.
158
QUADRICS.
[chap. III.
Let L be the plane \ifi + X«fa + ... + X^fr = 0.
Then it is proved in Salmon's Higher Algd>ra, Lesson v. and elsewhere
that, when A == 0,
But we may write by § 83 (3)
Hence, (Of $i)' = 0, reduces to, %/3\ = 0 ; that is to the condition that L pass
through the vertex. But this is the property of all polars and not merely of
tangential polars. Thus in this particular case of conical quadrics the
reciprocal equation to the ordinary point-equation, deduced as in the
general case, merely defines the vertex of the quadric.
(2) In order to find the nature of the condition which L must satisfy in
order to be a tangential polar of the conical quadric, suppose that ^ has been
chosen to be the vertex of the quadric.
Then (a$a7)» = a«if,« + 2a«f,f3 + ... = 0
is the equation of the quadric.
And («$^$^0 = f» (««?.' + «»?.' + ...)
+ f.(a«f/ + a„f/+...) + etc.
Hence, as before, the conditions that Z, which is the locus defined by
should touch the quadric are Xi = 0, and
Oa%» + 2a„'X,X,+ ... = 0;
where OL^', OL^y ... are the minors of Os^ Ou, ... in the determinant
ttjif, ttgy, ••• a^if
Let this determinant be called A^
The first condition, Xi == 0, is the condition given by the ordinary reciprocal
equation, namely that the polar should pass through the vertex.
Let the second condition be called the conical reciprocal equation.
(3) Now if we transform to any co-ordinate elements whatever, so that
any point h is the vertex, these equations become, 2/3X=sO, which is the
condition that b should be the vertex ; and {OC$Ly s 0, where the coefficients
Otn, dny ... satisfy the condition
flu* Amj ... Otii'>i=0.
84] RECIPROCAL EQUATIONS AND CONICAL QUADRICS. 159
In accordance with the notation explained § 78 (6) this determinant will
be called Ai.
(4) Suppose now that we are simply given the equation,
(a$i)»=o.
We have to determine what it is to be conceived as denoting when the
above determinant vanishes.
If we had the two equations
(O$X)« = 0, and 2^ = 0,
a conical quadric of vertex b would be determined. Hence it is possible to
conceive (Ot][X)* = 0 as denoting a conical quadric with an undetermined
vertex. This, however, is not satisfactory.
Let the reciprocal point-equation be formed. It follows from the
previous investigation that this equation is
where lOtu, i^u, ... are the minors of Otn, Gtu, Otu, ... in the determinant A^
This locus is two coincident planes forming a quadric.
If we choose i^ — 1 co-ordinate elements in this region and any co-ordinate
element Si outside it, then ((t$^Ly = 0 takes the form
7aV + 27„X,X, + ... =0.
Hence it is best to consider, (Ot$i)* = 0, as denoting in the reciprocal
point-form the region,
taken twice over, and as denoting in the original plane-form a quadric sur-
face of 1^ — 3 dimensions lying entirely within this region.
(5) If any vertex b be assumed and all the one dimensional regions
joining b to elements of this quadric of i/ — 3 dimensions be drawn, then a
conical quadric of i/ — 2 dimensions is obtained.
The vertex b should not be in the region,
i«u?i + iflu?2 + etc. = 0,
which contains the quadric surface of i^ — 3 dimensions, if a true conical
quadric is to be obtained. Hence we may call the region,
the ' non-vertical region '. Also call the quadric of i/ — 3 dimensions lying in
it the 'contained quadric'.
(6) Accordingly, summing up, given the equation, (tt$Xy = 0, where
A| = 0, we derive the reciprocal equation
A,f, + iai,f, + etc. = 0.
160 QUADRICS. [chap. III.
These two equations taken together represent a non-vertical region and a
contained quadric. This may be considered as the degenerate form of a
quadric defined by either of its two reciprocal equations.
Accordingly the equation, (OCj^Ly = 0 (when Ai = 0), gives the condition
to be satisfied by the co-ordinates of all regions of y — 2 dimensions whose
iuteisection with the non-vertical region is a tangential polar of the contained
quadric.
(7) Let us consider as a special case of the above investigations Geometry
of two dimensions. Let Ci, Ciye^he the co-ordinate points, and Ei, E^t E^ the
corresponding symbols denoting the straight lines e^y e^, eje^.
Then any point can be denoted by
and any straight line L by the equation
Also the equation,
denotes either that x lies on L or that L passes through x.
Any quadric (a$a?)' = 0 is a conic ; the determinant A is
A conical quadric, for which A = 0, is two straight lines. The reciprocal
equation is the tangential equation.
Conversely given the tangential equation (fl$Z)' = 0, the point-equation
is formed from it by the same law. Also if Ai = 0, then
(O$i)' = 0
splits up into two factors.
In this case let
Let p be the point
and let r be the point
Then {0O^Ly==0 is the condition to be satisfied by all lines which pass
through either j> or r.
Also flu = Wi/h, eta ; and fln = ^ (ps^s + p^f^t), eta
Hence Ai = «afl» - fl«" = - i (p,isr, - p^w^f,
and iflij = flssflsi — flfflflw = — i (pa^i - Ps^i) (pi^i — p\ «^8\
and jflis = — i (/>,«r, - pjWj) (p^ «r, — />,«•,).
84] RECIPROCAL EQUATIONS AND CONICAL QUADRICS. 161
Accordingly the non-vertical region is the straight line
(/>»«■, - pa^Tj) fi + (p,«ri - P1W3) f J 4- (pifsr^ - pj«r,) f, = 0.
This is the straight line joining the points p and q. Thus the non-vertical
region is a straight line, and the contained quadric is two points in it.
This agrees with Cayley's statements in his ' Sixth Memoir on Quantics*/
respecting conies in two dimensions.
Note. For further information in regard to what is known of the projective geometry
of many dimensions, cf. Veronese's treatise, Fondamenti di geometna (Padova, 1891),
translated into German mider the title, Qrwndeuge der Oeometrie von mehreren Dimensionen
und mekreren Arten gradliniger EinheUen in demerUcvrer Form entvnckdt (Leipzig, 1894).
* Cf. PhiU Tratu. 1859 and Collected MathemaHeal Papers^ vol. n., no. 158.
W, 11
CHAPTER IV.
Intensity.
86. Defining Equation of Intensity. (1) Let a complete region of
i^ — 1 dimensions be defined by the units Ci, e,, ... e„. Then by hypothesis
the intensity of the element represented by a^e^ is Op, since the intensity
of &p is by definition unity. But no principle has as yet been laid down
whereby the intensity of a derived element a can be determined ; where
a = ajCj + 0362 + ... + a^ey = Xae, say.
(2) Let a be the intensity, then we assume that a is some function of
Hence we can write
Now the intensity of /ui is /ta ; therefore we have the condition
/(/LUJti, /io,, ... fjLay) = fif{au Oa, ... a^).
Accordingly /(«!, Oj, ... a„) must be a homogeneous function of the first
degree.
(3) If a be at unit intensity, then the coefficients must satisfy the
equation
/(«!, 02, ... aM) = l (A).
This equation will be called ' the defining equation ' ; since it defines the
unit intensities of elements of the region. It will be noticed that the
equation does not in any way limit the ratios
which determine the character, or position, of the element represented by a.
Furthermore this equation essentially refers to the v special units 61, e^, ... e^
which have been chosen as defining (or co-ordinate) units of the region.
(4) Let <I>k{cIi, ... a,,) denote a rational integral homogeneous function of
the Xth degree, and <f>ii(oti, ... a„) a rational integral homogeneous function of
the fiih degree. Then the most general algebraic form of /(a,, ... a^) is
2 I^(?LllL?!l)U'-M
86] DEFINING EQUATION OF INTENSITY. 16S
If there be only one term in this expression, then equation (A) can be written
in the form
<l>K{^iy ««» ••• «•') = ^*»(«i> «fl> ••• <^) (-^y
(5) Let Oi c^ ... c,, be any other group of independent letters which can
be chosen as co-ordinates of the region. Let
a = 7i Ci + 7i Cj + - . + 7»' Cr = 27c.
Then the 7's are homogeneous linear functions of the a's.
Hence the defining equation (A') with reference to any other co-ordinate
elements becomes
'^a(7i> 7«» ••• 7.') = '^**(7i» 7s> ••• y^)\
where y^\ and y^f^ are homogeneous functions of the Xth and fith degrees
respectively.
(6) Again, when a,, a,, ... a^ all simultaneously vanish, the intensity is
unity when Oi = 1. But in this case equation (A') reduces to an equation
of the form
Therefore for this to be satisfied by ai = 1, we must have fi = 171. So the
coefficients of the highest powers of ai, a,, ... a„ on the two sides of the
equation are respectively equal.
(7) In the subsequent work, unless otherwise stated, we will assume the
defining equation to be of the form
where ^^(a^, a,, ... a,) is a rational integral homogeneous function of the fith
degree of the form
tti'' + ««'*+ ... + a/ + 2/)a/» «,'« ... a/p,
p being an arbitrary coefficient and
Raising each side of the defining equation to the /ith power it becomes.
Coefficients which satisfy the defining equation will be called the co-
ordinates of an element. They define the element at unit intensity. The
co-ordinates of an element must be distinguished from the co-ordinate elements
of a region, which have been defined before [cf. § 64 (8)].
86. Locus OF Zero Intensity. (1) It is obvious that there is oue
locus of V — 2 dimensions with exceptional properties in regard to the
intensities of its elements. For the equation
<l>^(ai, ... a„) = 0,
is a relation between the ratios Oq/ai, Os/aj, ... a^/ai; and it, therefore,
determines a locus which is such that all the elements of. it are
U— 2
164 INTENSITY. [chap. IV.
necessarily at zero intensity, according to this mode of defining the in-
tensity, and yet do not themselves vanish, since the coeflScients of the co-
ordinate extraordinaries do not vanish separately. Therefore in relation to
the given definition of unit intensity, elements of this locus are all at zero
intensity.
(2) Some other law of intensity is necessarily required in the locus of
zero intensity, at least in idea as a possibility in order to prevent the intro-
duction of fallacious reasoning. For if two terms a and a' represent the same
element at the same iuteusity, then a = a\ and the coefficients of the co-
ordinate elements in a and a' are respectively equal in pairs. But if the
element be in the locus of zero intensity a and pa are both zero intensity
according to the old definition. Hence from the above argument a = pa, and
therefore (/> — l)a = 0; and since p — 1 is not zero, a = 0, which is untrue.
Therefore in the locus of zero intensity in order to preserve generality of
expression some other definition of intensity is to be substituted, at least in
idea if not actually formulated. An analogy to this property of points in the
locus of zero intensity is found in the fact that two zero forces at infinity are
not therefore identical in effects, and that for such forces another definition of
intensity is substituted, namely the moment of the force about any point,
or in other words the moment of the couple.
(3) If the properties of the region with respect to the intensity are to be
assumed to be continuous, at any point of the locus of zero intensity one or
more of the co-ordinates must be infinite.
For the equation
viewed as an equation couneoting the absolute magnitudes of the co-ordinates,
can only be satisfied simultaneously with
viewed as an equation connecting the ratios of the co-ordinates if one or more
of the a*8 are infinite. In this case we can write the first equation in the form
Then if a^ becomes infinite, the equation,
VOp ftp/
between the ratios of the co-ordinates is simultaneously satisfied.
87. Plane Locus of Zero Intensity. (1) There are two special cases
of great importance, one when the locus of zero intensity is plane, the other
when it is a quadria
87] PLANE LOCUS OF ZERO INTENSITY. 165
C!oiisidering the case of a plane locus, by a proper choice of the unit
intensities of the co-ordinate elements of the complete region the equation of
the locus can be written in the form
fi + f, + ... + f. = 0.
(2) Let ^1, «£> ... e„ be these co-ordinate elements, and let Oi, Os, ... a,, be
another set of independent elements at unit intensity to be used as a new set
of co-ordinate elements.
Let Oi = aii6i -h ttM^a + . . . + ai„6v,
Then by hypothesis au + an 4- ... + oiu= 1,
with v — 1 other equations of the same type.
Let any element x at unit intensity be given by 2fe and also by 2i;a.
Then ^1 + ^2+ ... + ^..= 1.
But by comparison fi = aui/i -h Oai/a + . . . + ^Ki^ir,
with V — 1 other equations of the same type.
Hence substituting for the ^'s in the defining equation, we get
(flfu + Otia 4" . . . + fltir) ^1 4" (fltn 4- Claa 4" . . . 4- (X^ '^2'^ ••• — !•
Therefore using the defining equations for the a's, there results
% + ^2 4- . . . 4- 171' = 1>
as the defining equation for the new co-ordinates. It follows that if this
special type of defining equation of the first degree hold for one set of co-
ordinate elements it holds for all sets of co-ordinate elements.
(3) Any one-dimensional region meets the locus of zero intensity in one
element only, unless it lies wholly in the locus. Let a and h be two elements
at unit intensity defining a one-dimensional region. Then by subsection (2)
the intensity of any element fa 4- 176 in the region 06 is f 4- 17. Hence h — a
is of zero intensity. Let 6 — a = w ; then u is the only element in 06 at zero
intensity according to the original law of intensity, but possessing a finite
intensity according to some new definition.
(4) If p — 1 of the co-ordinate elements, where /> < v, be assumed in the
region of zero intensity and the remaining i^ — /> 4- 1 outside that region, then
the defining equation takes a peculiar form. For let t^i, ti^, . .. u^i be the co-
ordinate elements in the region of zero intensity , and 6p, 6p+i, ... 6„ the remain-
ing co-ordinate elements. Then any element can be expressed in the form
XKu 4- 2f «.
166 INTENSITY. [chap. IV.
Now any element of the form SKu lies in the region of zero intensity. Hence
the defining equation must take the form
ip + fp+i 4- . . . 4" f r = !•
li p = v, then the co-ordinate elements t^, ... i£^_i completely define the
region of zero intensity.
Let e denote the remaining co-ordinate element. Any element can be
written X\u -h ^e. The defining equation becomes, f = !•
88. QuADRic Locus OF Zebo Intensity. (1) Let the intensity of the
point a? be -h {(a$^)*}*
Then the locus of zero intensity is the quadric sur&ce (a$^)* = 0.
(2) Let us assume this quadric to be closed, or imaginary with real
coefficients [cf § 82 (4)]. If x lie within this quadric and y lie without it
(the quadric being real and closed), then by § 82 (11)
{al^xf and (a$y)»
are of opposite sign. Suppose for example that {oL$xf is positive for elements
within the quadric. And let
(a$a?)» = /i« and (al^)> = -i;»;
where fi and v are by hypothesis real, since the co-ordinates of x and y are
real. Then the intensities of x and y as denoted by the symbols x and y are
fL and V(— ^) respectively.
(8) The symbols which denote these points at unit intensity are xjfi, and
y/V(— i^). Hence although the element y is defined by real ratios, its co-
ordinates at unit intensity are imaginaries of the form ti/i, 1172, •••1 where
171, i/a ... are real.
Such elements will be called 'intensively imaginary dements ' If the
element be defined by real co-ordinates, its intensity is imaginary. Those
elements such that real co-ordinates define a' real intensity will be called
' intensively real elements.*
(4) If intensively real elements lie without the quadric of zero intensity,
then intensively imaginary elements lie within it, and conversely. It is to
be noted that both sets of elements are real in the sense that the ratios of
their co-ordinates are real.
(5) If the quadric of zero intensity be imaginary, then all real elements
are intensively real.
89. Antipodal elements and opposite intensiiies. (1) Since
(«$^)" = («$-^y>
the intensities of x and — x are both positive and equal, when the locus of zero
intensity is a quadric. An exception, therefore, arises to the law that if S^
and X^'e denote the same element at the same intensity, then
90] ANTIPODAL ELEMENTS AND OPPOSITE INTENSITIES. 167
Let the generality of this law be saved by considering the intensities denoted
by a and — x, though numerically the same, to diflfer by another quality which
we will call oppositeness.
(2) Another method of evading this exception to the general law is to
regard x and — a; as two different elements at the same intensity. This is
really a special case of the supposition alluded to in § 65 (9). Let x and — x
be called antipodal elements.
Li this method the quality of oppositeness has been assigned to the
intrinsic nature of the element denoted, whereas in the first method it was
assigned to the intensity. When the quadric locus of zero intensity is real
and closed, the first method is most convenicDt ; when it is imaginary, either
method can be chosen.
(3) Antipodal elements have special properties.
If any locus include an element, it also includes its antipodal element.
If two one-dimensional regions intersect, they also intersect in the anti-
podal element. Hence two one-dimensional regions, if they intersect, intersect
in two antipodal points.
A one-dimensional region meets a quadric in four points, real or imaginary,
namely in two pairs of antipodal points.
(4) The sign of congruence, namely = [cf. § 64 (2)], connects symbols
representing antipodal points as well as symbols representing the same point.
90. The Intercept between two elements. (1) The one-dimensional
region which includes ei and $2 may be conceived as divided by the elements
ei^B into two or more intercepts. For the element fi6i 4- ^2^ maybe conceived
as traversing one real portion of the region &om 6i to ^2, if it takes all
positions expressed by the continuous variation of ^s/fi from 0 to + oo .
Similarly it may travel from «i to e, through the remaining real portion of
the region by assuming all the positions expressed by the continuous variation
of f Vf 1 fro^ 0 to — 00 .
A one-dimensional region may, therefore, be considered as unbounded and
as returning into itself [cf. § 65 (9)].
(2) Assume that the expression for the intensity is linear and of the
form 2f. Then the locus of zero intensity cuts the region ei^g at the element
defined by f^fi = — 1.
Hence as fa/f i varies continuously from 0 to -h oo , the element x does not
pass through the locus of zero intensity, and its intensity cannot change sign,
if fa aiid f 1 do not change sign.
Let this portion of the region be called the intercept between ei and eg.
Also let the other portion of the region be called external to the portion
limited by e^ and e^, which is the intercept.
168 INTENSITY. [chap. IV.
(3) An element on the intercept between ei and ^ will be said to lie
between e^ and e^*
Also the external portion of the region is divided into two parts by the
element of zero intensity, e^ — ei. Let the continuous portion bounded by
61 and e^ — ei and not including e^ be called the portion beyond 61, and let the
portion bounded by e^ and eg — ^i and not including ei be called the portion
beyond 6a.
(4) Assume the intensity to be {(aja:)^} .
Let the locus of zero intensity be the real closed quadric, (aja?)* = 0.
Firstly, assume that ± x denote the same element at opposite intensities.
Let the two elements e^ and e^ both belong to the intensively real part of
the region. Then x moves from e^ to e,, as f^/fi varies from 0 to 00 or from
0 to — 00 . Now [cf. § 82 (9)] since (a$ei)' and {o^e^ are both of the same
sign, as x moves from e^ to e^ by either route it must either cut the surface
of zero intensity twice or not at all. Call the latter route the intercept
between ei and e^. The intercept only contains intensively real elements.
(5) If the quadric («$«?)* =0 be imaginary, then the one-Klimensional
region e^e^ does not cut it at all in real points. Hence there is no funda-
mental distinction between the two routes from ei to e^, and both of them
may be called intercepts between e^ and e^. Also all real elements are inten-
sively real.
Hence a one-dimensional region is to be conceived as a closed region, such
that two elements e^e^ divide it into two parts.
(6) Secondly, assume that ± x denote two antipodal elements.
Assume that the quadric (aja?)* = 0 is entirely imaginary. The two routes
from ei to ea are discriminated by the fact that the one contains both antipodal
points — ei and — ^2, and the other contains neither. Let the latter portion
of the region be called the intercept, and the former portion the antipodal
intercept.
The case when (a$a?)* = 0 is real is of no practical importance, and need not
be discussed.
Note. Graasmann does not consider the general question of the comparison of
intensities. In the Ausdehnungdehre von 1844, 2nd Part, Chapter i., §§ 94 — 100, he
assumes in effect a linear defining equation without considering any other possibility.
In the Aufdeknungdehre von 1862 no general discussion of the subject is given ; but in
Chapter v., * Applications to Geometry,' a linear defining equation for points is in efifect
assumed, and a quadric defining equation for vectors — assumptions which are obvious and
necessary in Euclidean Geometry. It should also be mentioned that the general idea of
a defining equation, different for different manifolds, and the idea of a locus of zero
intensity do not occur in either of these works. Also v. Helmholtz in his Handimch der
Physwlogischen Optik, § 20, pp. 327 to 330 (2nd Edition) apparently assumes that only a
linear defining equation is possibla
BOOK IV.
CALCULUS OF EXTENSION.
i
CHAPTER I.
Combinatorial Multiplication.
91. Introductory. (1) The preceding book has developed the general
theory of addition for algebras of the numerical genus (c£ § 22). The first
special algebra to be discussed is Orassmann's Calculus of Extension* This
algebra requires for its interpretation a complete algebraic system of mani-
folds (cf. § 20). The manifold of the first order is a positional manifold of
v — 1 dimensions, where v is any assigned integer ; the successive manifolds
of the second and higher orders are also positioual manifolds (cf. § 22) ; the
manifold of the i/th order reduces to a single element ; the manifold of the
(i/ + l)th order is identical with that of the first order. Hence (cf. §20),
when the manifold of the first order is of v — 1 dimensions, the algebra is of
the i/th species.
(2) It follows £rom the general equation for multiplication of algebras
of the numerical genus given in §22, that if two points a{=1ae) and
6(=2)8c) be multiplied together, where ^i, es ... 6,, are any p reference points,
then
ab = Sew . Ifie = S2 (op/Si^ep^^);
where |j = 1, 2 ... v, and a^fia are multiplied together according to the rules of
ordinary algebra.
(3) Thus the products two together of the reference elements ei, 6, ... 6,
yield i^ new elements of the form (eiBi), (e^), (fiie^, (e^), etc. These v*
elements (which may not all be independent) are conceived as defining a
fresh positional manifold of i/' — I dimensions at most, and a6 is an element of
this manifold. This is the most general conception possible of a relation
between any two elements of a positional manifold which may be symbolized
by a multiplication.
(4) No necessary connection exists between the symbols (jSiei), {Cie^,
(«a)> (^sAX etc.: they may therefore, as far as the logic of the formal
symbolism is concerned, be conceived as given independent reference elements
* Cf. Die AuBdehnungtlekre von 1844, and Die Atudehnungalekre von 1862, both by H.
Grassmann.
172 COMBINATORIAL MULTIPLICATION, [CHAP. L
of a new positional manifold. But on the other hand we are equally at
liberty to assume that some addition equations exist between these i^
products, whereby the number of them, which can be assumed as forming
a complete set of independent elements, is reduced. These products of
elements are then interpreted as symbolizing relations between the elements
of the manifold of the first order which form the factors ; and thus the mani-
folds of orders higher than the first represent properties of the manifold of
the first ordei: which it possesses in addition to its properties as a positional
manifold. Let any addition equations which exist between products of the
reference elements ei, ^s . .. &„ be called * equations of condition ' of that type
of multiplication which is under consideration.
92. Invariant Equations of Condition. (1) The equations of con-
dition will be called invariant, when the same equations of condition hold
whatever set of v independent reference elements be chosen in the manifold
of the first order*.
(2) For products of two elements of the first order, there are only two
types of multiplication with invariant equations of condition, namely that
type for which the equations of condition are of the form
(cA) + (ea^p) = 0, (6pCp) = 0 (1);
and that type for which the equations of condition are of the form
(«A) = (6a€p) ....(2).
For assume an equation of condition of the most general form possible,
namely
au(eiei) + ai2(ei«B) + «2i(«2ei) + --=0 (8).
Then if aci, x^-.-os^ be any v independent elements, this equation (3) is to
persist unchanged when iCi,X2...x^ are respectively substituted for ei, 6^ ...6„.
Thus in equation (3) change ei into f^i, where f is any arbitrary number,
not unity. Subtract equation (3) from this modified form, and divide by
f — 1, which by hypothesis is not zero. Then
p
Hence since ^ is arbitrary,
au(«iei) = 0, 2 {a,p(ei6p) + api(ep6i)l = 0 (4).
Therefore by hjrpothesis these forms are to be invariant equations of condition.
Hence the second of equations (4) must still hold when ^$2 is substituted
* The type of maltiplioation is then called by Orasemaim (of. AuadeknungtUhre von 1862, § 50)
' linear.' Bnt this nomenclatore clashea with the generally accepted meaning of a ' linear algebra *
as defined by B. Peiroe in his paper on Linear Associative Algebra, American Journal of Mathe-
maticit vol. xy. (1881), The theorem of subsection (2) is due to Ghnssmann, of. loc. eit.
93] INVABIANT EQUATIONS OF CONDITION. 178
for e^, f being any number not unity. Thus, as before, by subtraction and
division by f — 1,
«« i^^) + Oai (e^) = 0.
Since this equation is invariant, it must hold when ei and e^ are interchanged,
thus by subtraction,
(ttia - Oa) {(«i^) - (e^)] = 0.
(8) Firstly assume, Uu^a^. Then, if ei and Ca are any two of the
reference elements,
(eie2)-\-ie^)=-0 (5).
Now since this equation must be invariant, put e^ + ^ei for e^, where f is any
number not unity ; then by subtraction we find the typical form
(^1^ = 0 (6),
and this satisfies the first of equations (4).
It is evident and is formally proved in § 93 (3) that equations of condition,
of which equations (5) and (6) are t3rpical forms, actually are invariant.
(4) Secondly assume, (€162)^(6^1), as the t3rpical form of equation of
condition. Then it is immediately evident that (xy) = (yx), where x stands
for 2fe and y for ^e: Thus this form of equation of condition is invariant.
Also substituting x instead of 61 in the first of equations (4), which is
invariant, it is obvious that au'^O. Hence this equation introduces no
further equation of condition.
Thus there are only two types of multiplication of two elements of the
manifold of the first order which have invariant equations of condition.
93. Principles of Combinatorial Multiplication. (1) Let the
multiplication be called ' combinatorial ' when the following relations hold :
(W = (ea«a)=...=(«rO = 0j ^ ^'
(2) The second of equations (1) follows fi-om the first, if the first
equation be understood to hold in the case when /> = <r. For then (epiBe) = (Cpp),
and therefore (6p6^) 4- (eaep) = 2 (epCfi) = 0.
(3) Equations (1) and (2) as they stand apply to one given set of
independent elements, ^i, e2...e„. Now if a = Xae and b==l,fie, then the
product ab takes the form 2 (a^/S^ - a^fip) {%e^ ; and the number of indepen-
dent reference elements of the type e^e^ in the new manifold of the second
order created by the products of the reference elements of the first order is
^1/ (1^ — 1). Similarly the product ha becomes 2 (a,ri8p - ^^ i^ffiv)-
Thus for any two elements a and h of the manifold of the first order
equations of the same t3rpe as equation (1) hold, namely
(a6) + (6a) = 0, (aa)«(66) = 0.
174 COMBINATORIAL MULTIPLICATION. [CHAP. I.
(4) Equation (2) expresses what is known as the associative law of-
multiplication. It has been defined to hold for products of t^e v independent
elements ei, es ... e^. It follows from this law and from equation (1) that
€162 . . . ^p^p+l • •"• ^<r — V^l^ • • • ^p-l) V^p^p+l) (^p+9 • • • ^a)
= — (€162 . . . Cp-i) (^p+l^p) (fift+2 • • • ^<r)
= {€162 . . . 6p— i^p+i^p^p+a • • • ^v)'
Accordingly any two adjacent factors may be interchanged, if the sign of the
whole product be changed.
By a continually repeated interchange of adjacent factors any two fisuitors
can be interchanged if the sign of the whole product be changed.
Again, if the same element appear twice among the factors of a product,
the product is null. For by interchanges of factors the product can be
written in the form («i«i. ej^...), where ei is the repeated fiactor. But
(^^) = 0. Hence by § 21 (616162^. . .) = 0. It follows from this that in a region of
V — 1 dimensions products of more than p &ctors following this combinatorial
law are necessarily null, for one factor at least must be repeated.
(5) It remains to extend the associative law and the deductions from it
in the previous subsection to products of any elements. In the left-hand
side of equation (2) let any element, say 6p, be replaced by ep' = 6p + dex,
where e is any arbitrary number.
Then by the distributive law of multiplication (cf. § 19),
{Bi6i . . . 6p )yep+i6p^ • . . 60^) = (fii^i • • . ep){6p+ieft^ . . . ear) + cf (fie^ . . . Bk) (^p+i^p+a • • • ^<r)
^ V^l^ • • • ^p^p+l^p+a • • • B<r) * ^ v^l^ • • • ^X^p+l^p+9 • • • ^<r/
= (- Vjr^iBf, . Ciea ... Cp+iCp+a ... 6<r) + (- Vjr^d{eK . eiea ... Vi^P+a ••• O
= (— l/*~ \fip . ^^ • • . 6p4.i6p4.s . . . 60.).
Also by a similar proof
^^ ••» Bp 6p-|-i6p^.9 ... Bff *5 ^~- M.j^ \Bp . B^B^ ••• Bpj^\Bp^ ... B^^)*
Xience KBiB^ • • • Bp J yBp^xBp^^ • • • Bff) ^ BiB^ » • » Bp Bp^iBp^% ... B^*
Hence the associative law holds when Bp has been modified into Bp'; and by
successive modifications of this type ^,^...6„ can be modified into ai,CLi.,.at^,
where Oi, Os ... a„ are any independent elements.
(6) The only deduction in subsection (4) requiring further proof to
extend it to any product is the last, that in a region of 1/ — 1 dimensions
products of more that v fiax^tors are necessarily null, lliis theorem is a
special instance of the more general theorem, that products of elements
which are not independent are necessarily null. For let Oj, Os.-.ap be in-
dependent, but let ttp+i = ttiOi 4- . .. + OLpap,
Then (fli^ • • • ^pO^p+i) = ^1 (<h(h • • • (^p<h) + >»• +OLp (oiOs . . . af/if^.
94] PRINCIPLES OF COMBINATORIAL MULTIPLICATION. 176
Thus every product on the right-hand side has a repeated factor and is
therefore null
(7) Conversely, let it be assumed that a product formed by any number
of reference elements is not null, when no reference element is repeated as a
factor.
94. Derived Manifolds. (1) There are i/!/(i/-/>)!p! combinations
of the V independent elements ei, ^2 ... ^v taken p together (p < v). Let the
result of multiplying the p elements of any one combination together in any
arbitrary succession so as to form a product of the pth order be called a 'multi-
plicative combination * of the pih order of the elements ei, eg ... 6^.
There are obviously v\l{v — p)\ pi such multiplicative combinations of the
pth order.
(2) It is easy to prove formally that these multiplicative combinations
are independent elements of the derived manifold of the pth order (cf §§ 20
and 22).
For let ^1, J&a, ...-Cir, ... be these multiplicative combinations. Assume
that
Then if Ei denote the multiplicative combination (616^... 6p), the v — p ele-
ments ep+u e^+s ... 6y do not occur in Ei, and one at least of these elements
must occur in each of the other multiplicative combinationa
Now multiply the assumed equation successively by e^+i, ep+a...e^, then
by § 93 (4) all the terms become null, except the first term.
Accordingly fli (ejea . . . 6„) = 0.
But (ei^g.-.^r) is not zero by § 93 (7). So ai = 0. Similarly aa = 0, as=0,
and so on.
It follows that the sum of different multiplicative combinations cannot
itself be a multiplicative combination of the same set of reference elements.
(3) The complete derived manifold of the pth order is the positional
manifold defined by the v\l{v—p)\p\ independent multiplicative combinations
of the pth order formed out of the v reference elements of the first order.
Thus the manifold of the second order is defined by reference elements of the
type (Bf^eJ), of which there are ^v(v — l)y the manifold of the (i/-l)th
order is defined by reference elements of the type (6iCa-..^i^i)> of which
there are j/; the manifold of the vth order reduces to the single element
(4) The product of any number of independent elements of the first order
is never null, no factor being repeated.
For let Oi, Oa, ... a^, (a<v)he a independent elements of the first order.
Then by § 64, Prop. V. Corollary, v — o- other elements a^+u •••«•' can be
added to these elements, so as to form an independent system of v elements,
176 COMBINATORIAL MULTIPLICATION. [CHAP. L
Let V equations hold of the typical form
Then by § 93,
where A is the determinant X±auOL^... a^p.
Now A is not null, since all the elements di,dt,.,.dp are independent, and
(61 eg... ep) is not zero by § 93 (7). Hence (oiOa ... d,) cannot be null
(cf. § 21).
96. Extensive Magnitudes. (1) Consider a product of fi elements,
where fi<v: let all these elements, namely Oi, as...a^, be assumed to be
independent. Then they define a subregion of /a — 1 dimensions, which we
will call the subregion Af^. Let di\ o^^.. a/ be /i other independent elements
lying in the same subregion Afi. Then fi equations of the following type
must be satisfied, namely
di ^ Aqidl "T A|s|(Zj| + ... + ^^Itdfif
du, ^ AfujCll "T" AimjQq ^ ... "t" \ujijOuum
Hence by multiplication we find, remembering the law of interchange of
factors,
dlQ>i . . . a/ = A (did^ . . • dfi) \
where A stands for the determinant
♦Ml > ^18 > ••• ^l/*
A«2| , A22, .•• A<^
If the elements Oi', aa'...a/ be not independent, then (aj'a5,'...a^') = 0,
and A = 0 ; and hence in this case also
(didi . . . dj!) = A (did^ . . . dfi).
Thus if Oi', Oa' ... a/ and Oj, o^ ... a^ be respectively two sets of independent
elements defining the same subregion, then [cf. § 64 (2)]
(til d% .,. dfj) = (OiCZa . . . a^).
(2) Conversely, if (aj'a,,'... a/) = (aiaa...a^), where neither product is
zero, then di\ di ... a/ and Oi, Oa ... a^ define the same region : or in words,
two congruent products respectively define by their factors of the first order
the same subregion of the manifold of the first order.
For we may write (di'di ... dfi)=^\(dia^...df,). Multiply both sides
by Oj, then (a/o8'...a/ai) = 0. Hence by § 93 (6) and § 94 (4) Oj lies in
the region (oi', di ... df/). Similarly a^ lies in the same region, and so on.
Thus the two regions are identical
r
96] EXTENSIVE MAGNITUDES. 177
(3) A product of /x elements of the first order represents an element of
the derived manifold of the fith order (of. § 20) at a given intensity ; two
congruent but not equivalent products represent the same element but at
different intensities. Now an element of the manifold of the fith order,
which is represented by a product, may by means of subsections (1) and (2)
be identified with the subregion of the manifold of the first order defined by
that product. Thus the product is to be conceived as representing the
subregion at a given intensity. Then we shall, consistently with this con-
ception, use the symbol for a product, such as J.^ (where ^^ = 0103... a^),
also as the name of the subregion represented by the product.
(4) This symbolism and its interpretations can have no application
unless a subregion is more than a mere aggregate of its contained elements.
It is essentially assumed that a subregion can be treated as a whole and that
it possesses certain properties which are symbolized by the relations between
the elements of the derived manifold of the appropriate order. Thus a
subregion of the manifold of the first order, conceived as an element of a
positional manifold of a higher order, is the seat of an intensity and the
term which symbolizes it always symbolizes it as at a definite intensity.
(6) A positional manifold whose subregions possess this property will
be called an eoctensive manifold.
Let a product o{ p (p<v) elements of the first order (points) be called a
regional element of the pth order, and also a simple extensive magnitude of
the pth order.
Let regional elements of the first order be also called points, as was done
in Book III. : let regional elements of the second order be also called linear
elements or forces: let regional elements of the(i/ — l)th order be called
planar elements.
Also it will be convenient to understand 'regions' to mean regions of
the manifold of the first order, unless it is explained otherwise.
(6) Elements of the extensive manifold of the first order (Le. points)
will be denoted exclusively by small Koman letters. Elements of the
derived manifolds, when denoted by single letters, will be denoted exclusively
by capital Roman letters.
96. Simple and Compound Extensive MAONnuDES. (1) There is one
difficulty in this theory of derived manifolds which must be carefully noted.
For example let the original manifold be of three dimensions defined by
reference elements ei, e^y e^, 64. The reference linear elements of the manifold
of the second order are (cjea), (^^4), (^i^sX («a), (^^4), («a^)-
Then any element P of the positional manifold defined by these six
elements is expressed by
P = ttm (6i«si) + TTu (e^4^ 4- 7r„ (e^) + tt^ (e^*) + ^u (^^4) 4- Wa (^.
w, 12
178 COMBINATORIAL MULTIPLICATION. [CHAP. I.
But if an element of this manifold of the second order represent the
product of two elements 2^6 and S176 of the original manifold, it can be
expressed as
where (fpi7<r) stands for ^^ri^ - f^i/p.
But the following identity holds
(?i^2) (?,^4) + (f 117.) (&i7s) + (£174) (617.) = 0.
Accordingly P does not represent a product of elements of the original
manifold unless
'Wia'W84 + 7ri,7r42 + "WMTTaB = 0..
Thus only the elements lying on a quadric surfisM^e locus in the positional
manifold of five dimensions (which is the manifold of the second order)
represent products of elements of the original manifold.
(2) Let a derived manifold of the pth order be understood to denote
the complete positional manifold which is defined by the i/I/p! (i/ — p)! in-
dependent reference elements. Let those elements of this derived manifold
which can be represented as products of elements of the original manifold
be called * simple ' : let the other elements be called * compound.' Let the
term regional element [cf. § 96 (6)] be restricted to simple extensive magni-
tudes ; and let compound element^ be termed compound extensive magnitudes
or a system of regional elements. The latter term is used since every compound
element can be represented as a sum of simple elements. Thus an extensive
magnitude of the pth order is an element of the derived manifold of the pth
order, and may be either simple or compound.
(3) The associative law of multiplication identifies the product of two
simple elements {E^ and E^) of derived manifolds of the pth and olih orders
{p-\-<T<v) with the simple element of the derived manifold of the (/> + o-)th
order formed by multiplying in any succession the elements of the original
manifold which are the factors of E^, and E„.
Thus the product of any two elements, simple or compound, respectively
belonging to manifolds of the pth and <rth orders yield an element, simple or
compound, of the manifold of the {p + <r)th order. But the product of two
compound elements may be simple.
In the case of simple elements, Ef, and E^, the subregions J&p and E^ of
the original manifold may be said to be multiplied together.
97. Fundamental Propositions. Prop, L If fifp be an element (simple
or compound) of the derived manifold of the pth order, and if {aS^ = 0,
where a is a point [cf. § 95 (6)], then 8^ can be written in the form (a5p_i) ;
where /8fp_i is an element of the derived manifold of the (/> — l)th order.
For the reference elements of the original manifold may be assumed
to be 1/ independent elements a, 6, c... Let -4,, A^..,Bi, J5j... be the
97] FUNDAMENTAL PROPOSITIONS. 179
multiplicative combinations of the pth order formed out of these elements.
Let,Aj, A 2... be those which do contain a, and let J?i, J?2-*' be those which do
not contsdn a.
Then we may write
flfp = ttiili + 02^12 + . . . + fiiBi + yS^a + . . . •
But by hypothesis
(aSf,) = 0 = (aA^) = (aA^) = etc.
Hence multiplying the assumed equation by a we deduce
Now (aJBi), (aJBa), etc. are different multiplicative combinations of a, 6, c, etc.
of the (/) + l)th order. Hence they are independent, and by hypothesis
they do not vanish.
Accordingly the above equation requires ySj = 0 = /Sg = etc.
Hence 5p = a^A^ + a^2 + etc. = (a/S^-i).
Corollary. If (eiC^ ..,e^{a< p) be a simple element of the <7th order, and
if a equations hold of the type e^Sp = 0 (X = 1, 2 ... or), thien 8^ = (ei«2 . . . e^Sp^);
where 8p^ is an element of the {p — <7)th order.
Prop. n. If A denote a regional element of the <rth order, and B denote
a regional element of the pth order (p < <r) such that the subregion A contains
the subregion B, then A can be written (BC) ; where 0 is a regional element
of the (a - p)th. order. For let the subregion B be defined by the a inde-
pendent elements 01,02... a^. Then to these independent elements a — p
other independent elements Op+i, ap+2...cp<r can be added such that the a-
elements Oi, Oj ... a, define the region A. But
-4 = A (oiOi ... ttpttp+i . .. a^) = A (B&) = (JBC?) ;
where C stands for the product (ap+iap+g ... a^), and C=AC\
Corollory. It follows firom the two foregoing theorems that the com-
binatorial product of two subregions is zero if they possess one or more
elements in common.
If they possess no common subregion their product is the region which
contains them both.
Prop. III. If Af, and A^r be two regional elements of orders p and a
respectively, and if p + <7 = i/ + 7, then we can write -4p = (Cy J?p_y) and
A^= (GyBtr-^), where Cy is a regional element of the 7th order and J?p__y
and Btr-^ are regional elements of the (p — 7)th and (cr — 7)th orders.
For the subregions Ap and A^ must contain in common a subregion of at
least 7— 1 dimensions. Hence we are at liberty to assume the regional
element C!^ as a common factor both to Ap and A^-
X2— 2
180 COMBINATORIAL MULTIPLICATION. [CHAP. I.
Prop. IV. All the elements of the derived manifold of the (i; — l)th
order are simple. For let A and B be two simple elements of the (y — l)th
order. Then, since (i/— l) + (i' — l) = i' + (i' — 2), we may assume by the
previous proposition a regional element C the (y - 2)th order which is a
common &ctor of A and B.
Hence A = (aC), and B = (bC), where a and b are of the first order.
Thus A+B=^(a + b)a
But a+biB some element of the first order, call it c.
Hence A + B=»cC.
But cC is simple. Hence the sum of any number of simple elements of
the (i; — l)th order is a simple element.
Note. All the propositions of this chapter are substantially to be found in the
Atisdehnunffslehre von 1862. The application of Combinatorial Multiplication to the theoiy
of Determinants is investigated by K F. Scott, cf. A Treatise on the Theory of DetemUnobnUy
Cambridge, 1880. Terms obeying the combinatorial law of multiplication are called by him
'alternate numbers.'
CHAPTER II.
Regressive Multiplication.
98. Progressive and Regressive Multiplication. (1) According
to the laws of combinatorial multiplication just explained the product of
two extensive magnitudes Sp and 8^ respectively of the pth and crth order
must necessarily be null, if p + a be greater than v, where the original
manifold is of i; — 1 dimensions. Such products can therefore never occur,
since every term of any equation involving them would necessarily be null.
In the case p:\-a >v we are accordingly at liberty to assign a fresh law
of multiplication to be denoted by the symbol flfpflf^. Let multiplication
according to this new law (to be defined in § 100) be termed * regressive,' and
in contradisfcinction let combinatorial multiplication be called progressive.
Thus it p + cKv, the product Sf,8^ is formed according to the progressive
law already explained. Such products will be called progressive products.
If p + o- > j;, the product 8/^8^ will be formed according to the regressive law.
Such products will be called regressive products. If p + <7 = i;, the product
5p/8i, may be conceived indifferently as formed according to the progressive
or regressive law.
(2) In this last case it is to be noted that 8^8^ must necessarily be
of the form a(ei6s...6r), where ^i,69...6y are v independent reference
elements of the original manifold. Since therefore such products can only
represent a numerical multiple of a given product, we are at liberty to assume
them to be merely numerical.
Thus for example we may assume
{(BA ... 6„) = 1, and (8p8a) = « ;
where it is to be remembered that /> + <r = i^.
Let a product which is merely numerical be always enclosed in a bracket,
as thus (eieg...^,); other products will be enclosed in brackets where con-
venient, but numerical products invariably so.
99. Supplements. (1) Corresponding to any multiplicative combination
Ef^ of the fith order (/a<v) o{ the elements 6i, gj . . . e^, there exists [cf. § 65 (4)]
182 REORESSIVE MULTIPLICATION. [CHAP. 11.
a multiplicative combination ^^-^ of the {p — /x)th order which contains those
elements as factors which are omitted firom Efi. Let it be assumed that
(eiCa ... 6^)= + 1.
Hence {E^^E^^^i) = + (cj^ . . . e„) = ± 1.
We may notice that if E'y^i^ be any other multiplicative combination of the
(i/ - fi)t\i order, then {Ef,E\^^ = 0.
(2) The 'supplement' of any multiplicative combination E^ of the
reference elements e^ ^...e„' and of the /xth order is that multiplicative
combination Ey^,^ of the {v — fi^th order which contains those reference
elements omitted from i^V multiplied in such succession that
{Ey^Ey^ii) = 1.
Let the supplement of -ff^ be denoted by |j&^.
(3) Then if j&„_^ contain the same elements as \Ey, but multiplied in
any succession, jF^-** will be called the multiplicative combination supple-
mentary to Ei^,
Then since {Ei^E^^ii)— ±\, we see that \Et^ = (Ef^Ey^f,)Ey^^; where
(Ef^Ey^f^ is treated as a numerical multiplier of j&„_^.
The fundamental equations satisfied by \Ef^ are
(E^\E;)^1, and (E;\E^) = 0;
where ^/ is any multiplicative combination of the /iith order other than Ef^.
(4) The analogy of the above definitions leads us in the extreme cases
to define
K^i^s^s • • ^r) = 1> c^d 1 1 = (^^2 . . . €„).
Since (^ej ... c„) = 1, it follows from these definitions that, |1 = 1.
(5) Let the supplement of a sum of multiplicative combinations of a
given order be defined to be the sum of the supplements. This definition
is consistent with that of subsection (2), since [cf. § 94 (2)] the sum of
different multiplicative combinations is not a multiplicative combination of
the reference elements.
Thus |(^^ + i?/ + ...)=|^^ + |^/+...
Let this definition be assumed to apply to the special case where Ef^ is
repeated a timea
Thus |(j&^ + -S^+... to a term8) = |i?^+|J&^+... to a tenns = a|^,4.
Hence Ko^m) = « |-S>-
Now let /A^Vy and E^^^E^^Y. Then the above equation becomes | a = a.
Also finally |(ajF^ + a'^/ + etc.) = a|^^ + a'|^/ + etc.
(6) The symbol | may be considered as denoting an operation on the
terms following it. It will be called the operation of taking the supplement.
100] SUPPLEMENTS. 183
This operation* is distributive in reference to addition and also in reference
to the product of a numerical factor and an extensive magnitude. For
|(J.+B)=:|^ + |jB,and \{a.A)=:\a.\A.
(7) Let the symbol \\A denote the supplement of the supplement of A.
If il be an extensive magnitude of the fith order, then
\\A^(-iy(^^A.
For with the notation used above,
\Ef^= (Ef^Ey^ Et^f^y and \E^ft^(Ey^ft,Eft)Efi.
Hence from (5) 1 1 -ff^ = {EfiE,^f^ \ ^^-^ = {EfiE^^f^ (^E^^i^Eii) E^,
But iE^^B^ = (- 1)^ ^"^"^ {KE^y)\ and {E^E^) = ± 1.
Hence ||^^ = (- 1>* <'-'*> ^^.
But . A^ taEf,; and therefore \\A = (-l)^<'^'*>il.
(8) It must finally be noted that the supplement of an extensive
magnitude must be taken to refer to a definite set of reference elements of
the original manifold, and that it has no signification except in relation to
such a set.
(9) The following notation for the operation of the symbol | on products
will be observed. The symbol will be taken to operate on all succeeding
letters of a product up to the next dot ; thus a \ bed means that | (bed) is to
be multiplied into a ; and a\bc.d means that | (be) is to be multiplied into
a and d into this product. Also a second | will be taken to act as a dot in
limiting the operation of a former | : thus \A\B mecms that \B is multiplied
into \A, and it does not mean \(A \B).
Again, | placed before a bracket will be taken to act only on the magnitude
inside the bracket: thus \{AB)C means that C is multiplied into \(AB).
Johnson's notation with dots might be employed [cf. § 20] : thus \ab.c. .d
would mean that c is multiplied into \(ah) and d into this product.
100. Definition of Eegbessivb Multiplication. (1) If A^, and A^
be extensive magnitudes of the pth and o-th orders, where p-ho v; then
\Aft and |-4^ are extensive magnitudes of the (v — /o)th and (i; — <7)th orders,
and (v — p) + (i' — <7) < V. Hence |-4p and \A^ can be multiplied together
according to the progressive rule of multiplication, already explained.
Now the regressive product of A^ and A^ will be so defined that the
operation of taking the supplement may be distributive in reference to the
factors of a product.
(2) Let the regressive product ApA^ be defined to be an extensive
magnitude, such that its supplement is \Af, {A^.
In symbols, | ApA, =^\Ap\ A^.
184 REGRESSIVE MULTIPLICATION. [CHAP. II.
Since |-4p \A^ is of the (2i; — p — o-)th order, the regressive product Af,A,
is an extensive quantity of the (p + <r — j')th order.
(8) If p + <r = V, then Af^A^^ can be indifferently conceived either as
progressive or as regressive. For if E^, E/, etc. are the multiplicative com-
binations of the reference elements of the pth order, we can write
Af, = %apEp, and il<r = Sa,|^p.
Hence (^p^^) = (V^ + K^ + ^*^'>
since (E^ |^p) = 1, and (E^ \E;) = 0.
Also |ilp=Sap|^p and |il^ = 2a<r||^p = (- l>*<'^'*^a<r^p.
Hence ( \A^ |ila) = (- Vf'^'^^^ {apO^C i^p . ^p)+ apV( |^/. Ei) + etc.}.
Now (|^p.^p) = (-l)p<''-p>(^p|^p) = (-l>»<'^p>.
Thus finally
( I ilp I A^ = OpOa + flpV + etc. = I {opO, + ttpV + etc.} = | (Af,A^.
(4) Again if p + <r < v, the product Af^A^ is progressive and the product
\A^ \A„ is regressive.
But by definition of the regressive product |-4.p 1-4,, we have
|{|^t^.} = ||ilp||il..
Now ||ilp = (-l>»<'--p>ilp, and || J^ = (-!)'<•--<') il^.
Hence \ { |ilpl^^} = (- lyt-^-p^+'^-^^pila.
Therefore, taking the supplement of each side,
II { l^p \A^\ = (- l)pc-p)+<r(--a) I ^^^^.
Now |ilp 1-4^ is an extensive magnitude of the (i/ — /> — <7)th degree. Hence
II { \A^\A^\ = (- l)e-p-')(p+') |ilp|il^.
Also (" 1 ) ^^^"^ ^"•"^^ = (— 1 )p <''~p^ ■•■' <''~*^ .
Hence | iipil^ = I -4p | A^.
(5) Finally therefore in every case, whether the product A^A^ be pro-
gressive or regressive, we deduce | ApA^ = |-4.p \A^,
101. Pure and Mixed Products. (1) A product in which all the
multiplications indicated are all progressive or all regressive (as the case may
be) is called pure ; if the multiplications are all progressive the product is
called a pure progressive product ; if all. regressive, a pure regressive product.
Thus if -4, J?, C and D be extensive magnitudes, the product AB . CD is
a pure regressive product if the product of A and B is regressive, and that of
G and 2), and that of AB and CD.
(2) A product which is not pure is called ' mixed.' Thus if the product
of -4 and B is regressive, and that of C and D is progressive, then the product
AB • GB is mixed.
102] PURE AND MIXED PBODUCTS. 185
(3) A pure product is associative.
This proposition is true by definition, if the pure product be progressive.
If the product (P) of magnitudes A, B, C, etc. be a pure regressive pro-
duct, then the product of 1-4, \B, \C, etc. is a pure progressive product.
But this product is associative.
Hence |P = |^. |5|a... = |il |J?|0... .
Taking the supplements of both sides
||P = ||il.||5||a... = ||^||5||C...,
hence P^A.BC.^ABG...,
For instance, if the product AB . CD be pure, we may write
AB.CD = ABCJD.
A mixed product is not in general associative.
102. Rule of the Middle Factor. (1) We have now to give rules
for the identification of that extensive magnitude of the (p + <t — i;)th order
which is denoted by the regressive product ApA„. This will be accomplished
by the following theorems.
(2) Proposition A. Let -ffp, E^y Er be three multiplicative combinations
of the reference units of the pth, <rth, and rth orders respectively, and let
p + <r + T = I/.
To prove that E^JE^ . E^E^ = (EfJSJSr) E^.
It will be noticed that the products EfJS^ and EfJSr are progressive, while
the final product of E^^ and E^r is regressive ; and also that {E^^E^) is
either zero or ± 1.
Case I. Let {EfJSaEr) = 0. Then since in this product there are only
V &ctors of the first order, one of the v reference elements must be absent
in order that another one may be repeated.
Let ei be the absent element. Then e^ is contained neither in E^^^ nor
in Ef^^, Hence it is contained both in K^^,) and in \(EfJEr).
Therefore \(E^^ . E^^) = |(^A) l(^/>^r) = 0.
Hence E^^ . E^^ = 0 = {E^JEJEr) E^.
Case IL Let (E^^Er) - ± 1.
In this case no factor of the first order is repeated in the product
(e,e.e;).
Hence [cf. § 99 (3)] \E„ = (E^E^^) E^r, \ Er = (ErE^^) B^^.
Hence \{ErE^)=:\E,\E„^{EJEf^;){EJS^r) E^^.E^r (1).
But from §99 (3) \(ErE^) ^ (ErE^E;) E^ (2).
/
186 REGEESSIVE MULTIPLICATION. [CHAP. II.
Also {ErE^;) = (- 1)-(P+') {E^^r)s (KE^r) = (- l)^ (E^^Er),
and (EJE^E,) = (- 1)p<'+-0>+') (E^^^Er).
Therefore from equations (1) and (2), and remembering that {Ef^E^E^= ± 1,
it follows that - E^^.E^r = {E^JEr)E,.
(3) Proposition B. If Af,, A^ A^ be any simple extensive magnitudes of
the pth, <7th and rth orders respectively, suet thAt p + <r + t = i/, then
For let us assume that this formula holds for the case when the factors of
the first order of A^] A,, and Ar are composed out of a given set of v inde-
pendent elements aiya2.,.av. Th^n we shall show that the formula holds for
products formed out of the set Oi', Og ... «„, where ai' = Sa^a^ and replaces Oj.
Now Oi may occur in -4p, -4^, or Ar.
Firstly assume that it occurs in A„, Let A^ = Oiil^-i; and let AJ= a^A^^^,
Then A J is what A^ becomes when a^ is everywhere substituted for Oj, and
Af^ and A^ are unaltered by this substitution.
Thus AJ =^ oi^Ao^i = 2a^ (a^il^_i).
Hence AfJLJ . Afjir = 2a^ (Affiif^A^^i . AfAr)-
But each product of the type A^a^A^-i . Af,Ar is such that the factors of
the first degree in A^,, a^^,_i, and Ar are composed of the set of elements
a,,a^...a„ Accordingly by our assumption
ApflffiAff—i . AfyAr = (Af/if^^^iAr) Ap,
Hence Af^A^/ . A^^Ar = Sa^ (Af/i^^^iAr) Ap = (A^ . Sa^a^ . A„^iAr) A^
=={ApaiA^^iAr)Ai, = {ApAtrAr)Af,»
Secondly, it can be proved in exactly the same manner that if Oi occurs
in J.,, so that when di' is substituted for Oi, Ar becomes Ar and A^ and A^
are unaffected, then A^A^ . ApAr^{ApA^Ar'^ Ap.
Thirdly, assume that Oi occurs in Ap.
Let Oi be changed into Oi' = aiOi + agOa* aiid let 4p = (Oi-Ap^i), and
Ap = (Oi -a.p_i).
If Oa occur in -4p_i, then
Ap = tti (oiilp-O + Oa (Mp-i) = «! (oiAp^i) = ai-4p.
Accordingly -4p' is merely a multiple of Ap, and we deduce immediately that
ApA^ . ApAr = a^'ApA^ . ilpil^ = a* (ApA^Ar) Ap
^(A;A^r)Ap\ ;
If a, does not occur in Ap-i, suppose that it occurs in A„. Let A^ = a»i4o_i.
Then iip'ila = ai (oiAp^iOiA^^i) + Oa (c^p-iOa^ir-i)
» ai{ai^p_ia2^<r-i) ^ ^i-^p^v
And ApAr = aiApAr + ci2((hAp-^iAr).
102] RITLB OF THE MIDDLE FACTOa 187
Hence A^A^ . AJAr^diAf^^ . AfJLr + (iiCLiAfyA, . a^f^iAr,
Now Af^9 . ApAr = (AfAtfAr) Af,.
Therefore by substitution,
A.p A-a • Ap A-r = Oil \ApA.aA.r) [CLiA-p + OijfliAp^i]*
But (ApA„Ar) = tti (-4^4^-4^), and tti-ip + ajOailp-i = Ap\
Hence finally il/il. . A;A r = (-A/il^ilT) A;.
But by repeated substitutions for Oi, Oi', etc. of the type Oi'ssaiOi + OaO,,
Oi" = fii(h' -^ fi^s$ and so on, Oi is finally replaced by any arbitrary element
Thus if any element of the set Oi, a, ... a^ be replaced by an arbitrary
element, the formula still holds. Hence by successive substitution the v
elements Oi, Og . . . a, can be replaced by v other elements &i, &2 • • • K*
But if Epy E^, Er be simple magnitudes formed by products of the
reference elements Cj, ^...e^, the formula, EpE^,.EpEr^{EfJEJE^)Ep^ has
been proved to hold by proposition A. Therefore if Ap, A^, Ar be simple
magnitudes formed by products of any set of elements Oi, a, . . . a, the formula,
ApAa . Af^Ar = (Af^AoAr) Ap, holds.
(4) Corollary. It is easy to see that the formula still holds if A^, and A^
be compound. But it does not hold if Ap is compound.
Proposition B is the foundation of all the formulae in this algebra. The
following important formulae given by propositions C and D can be deduced
from it.
(6) Proposition 0. ApA^r . ApA^ = (ApA^Ar) Ap, when p + <r + t = 2i/.
In this case the products J.^, and ApAr are both regressive. Hence
the products \Ap \Aa and \Ap \Ar are both progressive, and
(i' — p) + (i' — o") + (i; — t) = V.
Hence by proposition B, \Ap lA^ . \Ap \Ar = (|-4p \At, \Ar) \Ap,
Therefore by taking supplements of both sides
(6) Proposition D. ApA^, . AoA^ = (Af^^fAr) A^,
and -4^T . A^A^ « {ApA^A^ A^;
where p + <r + t = i/ or 2i/.
These formulae follow immediately from propositions B and C.
188 REGRESSIVE MULTIPLICATION. [CHAP. II.
For AfJL^.A^Ar=^(rVjrAaAf,.A^Ar
^{^Vr{A^^r)A,
= {ApA^Ar) Atf,
Similarly for the other formula.
(7) These formulae may all be included in one rule, which we will call
the rule of the middle factor, given in the following proposition.
Proposition E. Let Af, and A^ be two simple extensive magnitudes of
the pth and <Tth order such that p + <7 = i; + 7. Then the regions A^ and A^
have a common region of at least 7 — 1 dimensions. Let (7y be this common
region. Then we may write (cf § 97 Prop. III.) -4.p = Bf,^ Cy, and A^ = GyB^^.
And it is easy from the foregoing propositions to prove that
Ap Ao^ = £p_y(7y . Aa^ = (Bp-^Aff) Gy
= Ap . GyBff^y = {ApB^—y) Gy,
These formulae embody the rule of the middle factor.
103. Extended Rule of the Middle Factor. (1) But this rule
in its present form is not very easily applicable in most cases. Thus sup-
pose that the complete manifold be of three dimensions, so that i^ = 4, and
let Ap^^pqr, and A, = 8t; where p, q, r, 8, t are elements of the complete
manifold. Then to find the product pqr . st, the rule directs us to find the
element x which the line 8t must have in common with the plane p^ and to
write either pqr in the form uvx or ^ in the form xz ; and then
pqr . 8t = uvx . 8t = {uv8t)x, and pqr . 8t=^pqr . xz = (pqrz)x.
But no rule has yet been given to express x in terms of ;?, g, r, «, t
This defect is remedied by the following proposition embodied in equa-
tions (1) and (2) of the next subsection, which we will call the ' extended
rule of the middle factor.'
(2) Proposition F. Let Ap and £, be simple extensive magnitudes of the
pth and <rth orders respectively, and let p + <r = 1/ + 7, where 7 must be less
than V. Let G^, G^, etc. denote the multiplicative combinations of the 7th
order which can be formed out of the factors of the first order of A p. Then
we may write
Ap = Ap-yO y =S ilp_y G y = QW,y
where A^-y, Af^y, etc. are extensive magnitudes of the (p — 7)th order.
Then according to the extended rule of the middle factor
^5, = (il2Ly5,)0y+(il,%5,)C« + etc (1).
Similarly let 2)y, L^, etc. be the multiplicative combinations of the 7th order
formed out of the factors of the first order of 5^. Then we may write
B^ = Dy^JLy = i)»5?-y = eta,
where -B^.y, 5y_y, etc. are extensive magnitudes of the (<r — 7)th order.
103] EXTENDED RULE OF THE MIDDLE FACTOR. 189
Then according to the extended rule of the middle factor
A,B, = {A,BfS.^)U^' + (^^.y) i)« + etc (2).
Equations (1) and (2) form the extended rule of the middle £Ekctor which
has now to be proved.
Let Oi, Oa ... af, be the p factors of Af,, and let 6i, &a ••• &<r be the a factors
of 5a.
Let v^p other elements ap+i, ap+2...ay be added to Oi.^.ap, so as to
form a set of v independent elements.
Then we may write B^, in the form
/Si^y + ABy + etc.;
where Bl?, B?, etc. are the multiplicative combinations of the o-th order of
the elements Oi, a, ... a^; and any number of the coefficients )8i, ^9, etc. may
be zero.
Also, remembering that (/> — 7) + o- = 1/, let the index-notation be so
arranged that B^ contains those a's which do not appear in -4^-y, and JB?
contains those which do not appear in -4^_y, and so on.
Then it may be noted that to every magnitude A^}y there corresponds a
magnitude J?i^\ but not necessarily conversely.
Furthermore it is obvious that when X 4= /^
Then A,B, = 2y8^ {A^^J"^) = 2^^ (^p^^y C^ji - Bi'*^).
Now G^^ must represent a subregion contained in the subregion J?lf*^ ; since
Q^^ is a product of 7 of the a's which do not appear in A^lyy and J?^^ is a
product of all those a's which do not appear in A^ly, Hence by the rule of
the middle fSeu^tor
A^}y&^^ . 5^^^ = Up^Jy5lr>) C^'-l
Also since A)l^lyB^^ = 0, we deduce"
^M^}yB^^) = 2)8x {AflyB^:^^) = A^}, . t0,B^^^ = (AJT^yB,).
Hence finaUy A^B, = S (AJT^yB,) Cj"' ;
which is the equation (1) of the enunciation. An exactly similar proof yields
equation (2).
(3) The following formulae are important special examples of this ex-
tended rule of the middle factor.
Let v = S, the complete manifold being therefore of two dimensions.
Then pq.r8 = (pr8)q'-(qr8)p = (pq8)r''(pqr)8 (3).
Let 1/ = 4, the complete manifold being therefore of three dimensions.
Then pqr . st = st.pqr = (pqrt)8 — (p^8)t
= (pqst) r ■\- (rpst) q + (qrst) p (4).
190 REaRESSIVE MULTIPLICATION. [CHAP. IL
And pqr .stu^^-- stu.pqr = (pqr8)tu + (pqru)8t + (pqrt)ri8
= (pstu) qr + (rstu) pq '\- (qstu) rp (5).
(4) Take the supplements of these formulae.
When i/ = 3, the supplement of a magnitude of the first order is a
magnitude of the second order. Let P, Q, R, 8 be magnitudes of the
second order such that P = \p, Q = |?» etc.
Then by taking the supplement of (3) we deduce
PQ.R8 = (PRS)Q'(QR8)P = {PQS)R-(PQR)8 (3').
Again let i/ = 4 ; then the supplement of a magnitude of the first order
is one of the third order.
Let P, Q, R, 8,.T, Uhe put for \p, \q, |r, |«, \t, \u; and let the supple-
ments of equations (4) and (5) be taken.
Then PQR ,8T=^8T. PQR = (PQRT) 8 - (PQR8) T
= (PQ8T)R + (RP8T)Q + (QR8T)P (4').
And
PQR. STU=-8TU, PQR = (PQR8)TU+(PQRU)8T + (PQRT) U8
==(P8TU)QR + {R8TU)PQ-h{Q8TU)RP (50-
In fact by taking supplements any formula involving magnitudes of the
first order is converted into one involving planar elements, i.e. magnitudes of
the (v — l)th order; where the complete manifold is of i/ - 1 dimensions.
104. Regressive Multiplication independent of Reference Ele-
ments. (1) The rule of the middle factor and the extended rule disclose
the fact that the regressive product of two magnitudes A and B is inde-
pendent of the special reference elements in the original manifold which were
chosen for defining the operation of taking the supplement. Accordingly
regressive multiplication is an operation independent of any special reference
elements or of their intensities, though such elements are used in its defini-
tion for the sake of simplicity. Also it is independent of the fact that the
product of the v reference elements was taken to be unity for simplicity of
explanation. Thus the product may be assumed to have any numerical
value A which may be convenient [cf. § 98 (2)].
It would have been possible to define regressive multiplication by means
of the rule of the middle factor. It would then have been necessary to
prove that it is a true multiplication, namely that it is distributive in
reference to addition.
. (2) It is useful to bear in mind the following summary of results re-
specting the multiplication of two regions Pp and P^, of the pth and <rth
orders respectively :
105] REGRESSIVE MULTIPLICATION INDEPENDENT OF REFERENCE ELEMENTS. 191
If p + cKv, then PftPt, is progressive and represents the containing
region [cf. § 65 (6)] of the two regions P^ and P^ ; unless Pp and P^ overlap,
and in this ease the progressive product P/tP^ is zero.
If p-\-(r>v, then Pf^P^r is regressive and represents the complete region
common both to P^, and P^ ; unless P^, and P^ overlap in a region of order
greater than p + a — v, and in this case PpP^ is zero.
I{ p + <r^v, then (PpPir) is a mere number and can be considered either
as progressive or regressive.
The only formulae which in practice it is necessary to retain in the
memory are the extended rule of the middle factor [cf. § 103] and propo-
sition G of § 105.
106. Proposition G. If Oi, a,... ap be p points in a region of v
dimensions (i; > p), and if JBi, J?j . .. JBp be p planar elements, then
-1
{(iidi • • • dft^ BiB^ . . . £|p) ^
{(hBi\ ((hBi)...((hBp)
For assume that the formula is true for the number p — l respectively of
points and of planar elements, to prove that it is true for the number
p ; where p<v.
Let A^ denote the minor of the element {a^B^) of the above determinant.
Now the product of the p+l regional elements of the />th order {aia^.,,a^,
£i, J?3 ... JSp is a pure regressive product and is therefore associative.
Hence (ch!^ ...dp. BiB^ . . . B^ = {(oiOs ,,,ap. Bi) B^B^ . . . B^
= (oijBi) (ogO, . . . ttp . B^z"- Bp) + (a»Bi) (oiOs . . . a^ . BJB^... Bp)
+ . . . + (ctpBi) (oiOs • • • ^p-i • -BgiSs • • • Bp),
But since the theorem holds for the number p — 1, a^s '^cip. B^B^ . . . J?p = An,
with p similar equations.
Hence
(OiOa... ttp. J?iB2...Bp) = (aiJ?i)Aii + (a2Bi)A2i+...+(ap5i)Api
(oiBi), (oiBg) ... {a^Bp)
(OaBiX {<hB^ ... {(hBp)
(apBi), {apB^) . . . {apBp)
But when /> = 2,
(oiOa . BiB^ = (sh(h • -Bi) B^
= {i<hBi) Oi - ((hBi) Oi] J?8
Therefore the theorem is true universally.
106] HULLER'S THEOKEH8. 193
where D,, D,', etc., are the maltiplicative combinations of the rth order
formed out of &i, i,, ... b^, Ci, Ci, ... c,. If t be less than both p and o-, some
of the multiplicative combinations D^, D,' , etc., contain only h's, some only
c's, and aome both i's and c's. If t be leas than p and greater than o-
(assuming p > a), then some of the D,'a contain only &'s, and some contain
both i'a and c's ; but none contain only c's. If t be greater than both p and
a, then all the D^'a must contain both &'s and c's.
(5) Let the products BC, AB, AG be progressive. Then
p + tr <v, K+p < V, K + a <if.
By the extended rule of the middle factor
ABC=SiABC^,)C,. ACB = t{AGB,-,)B,.
Hence if a relation of the form of equation (i) holds, no i),'s mu^t exist
which contain both 6's and c's. But this condition can only hold when t= 1.
Hence the condition is that
*: + /j + <7 = i/ + l.
Also, remembering that c^SO^-i =(- l>'Sc^Oi'i, = (- lyBC, equation
(ii) becomes
And
ABC = 'liABC'^li)c^, ACB=X{ACB'fix)b^ = {-\y"'-'*t{AB^fi^O)h^.
Hence A.BC = {-\yABG-k^i-\Y''-»ACB (lii).
y — a)
ly the
iv).
'oduct
194 REGRESSIVE MULtl PLICATION. [CHAP. 11.
. By the extended rule of the middle factor
Hence ACB = 'S^(A (7,_«) (7,+^_^ B.
Accordingly if a relation of the form of equation (i) holds, the D^'s
[c£ subsection (4)] must consist of two classes only, namely those composed
only of c% and those which contain all the 6's.
But this is only possible if
p=i.
In this case B is of the first order and will be written 6.
Then remembering that
GrbC^^ = (- 1)^ bC, hCr-, 0^-r^, = 60,
and that /c-\- a— v = t — 1, a- — T + l=y — ic,
equation (ii) takes the form
A.BC={- ly 2 (AbC,^) a + 2 (^a-Tf i) bC^,,
^i-iyABC-^i-iy-'ACB (v).
This is the required equation of the form of equation (i).
(8) Let the products BC and AB he regressive and the product AC he
progressive. This case can be deduced from subsection (7) by the method of
subsection (6).
The necessary condition for the existence of the required addition
gelation is
p= I' — 1.
Then from the assumptions it follows that
/C>1, 0->l, K-\-a'<P^ /C + I/ — 1 + 0' = I' + T.
Also A . BC = (-l)^ ABC -\- {-ly-^^ ACB (vi).
(9) Let the products BC and A Che progressive, and the product ABhe
regressive.
Now A.BC^^i-'iy^A.CB.
Hence this case can be deduced from that of subsection (7).
The necessary condition is that
o- = l.
Then A,CB = {^iy ACB + (- 1)^' ABC
Hence A .BC=-('- ly^-' ABC + ('-iy+^ACB (vii).
(10) Let the products BC and AC he regressive, and the product AB be
progressive.
Then from the previous subsection
0' = y — 1,
And .1 . 5(7= (- ly^i^-^^ ABC+ (- 1)«^^ ACB
= (-l>»^^MB(7 + (- ly+^ACB (viii).
107] MUJJiBR'S THEOREMS. 195
(11) Let the product BG be progressive, and the products AB and AC
be regressive.
Then AB = ^ (AB,.,) 5p+..„ AC^t (AC,) C^+,.,.
Hence ABC^ S (AB^,) 5p+«^, G, ACB^t (AG^) a+«-. B.
Thus the D/s of subsection (4) equation (ii) must either contain all the
6*8 or all the c's ; and thus the Dp+^^'s of the same equation must contain
only Vb or only c's. Hence the Dp+^r^r^ are of the first order, that is to say,
P + 0- — T=l. But it+p + o- = i/ + T.
Hence the required condition is that
« = y — 1.
Then AB = 2 (Ab^) fij^\, AC= (Ac^) G^l^ ;
where B^Tli b^ = B, C^l^ c^ = G
Thus ABC=X(Ab^)Bfi^2iG, AGB = t(Ac^)GlrliB.
Now JBil\C5^ = (-l)'£^2i6^C = (-l)'50,
and G^li Bc^ = (- 1>» Oil\ c^B = (- 1>» 05 = (- 1)^+^ BG
Hence by comparing with equation (ii) of subsection (4)
A.BG=(^l)'ABG + (-iy('-^^)AGB (ix).
(12) Let the product BG be regressive, and the products AB and AG be
progressive.
Then from the previous subsection the condition is
Also A . BC=(- ly-'' ABG+ (- i)i^-PH^-'+i) AGB (x).
(13) It has nowhere been assumed in the foregoing reasoning that A is
simple. Accordingly A may be compound.
107. Applications and Examples. (1) The condition that an ele-
ment X may lie in a subregion P^ of p — 1 dimensions is, a?P^ = 0. This
equation may therefore be regarded as the equation of the subregion.
(2) The supplementary equation is, \x jPp = 0. The product of jo; and
|Pp is regressive, and the equation indicates that \x and |Pp overlap in a
regional element of an order greater than the excess of the orders of \x
and |Pp above v. Now the order of \x is i/ — 1, and the order of |Pp is
v — p. Hence the order of the common region is greater than
(i/-l) + (i'-p)-i',
that is, is greater than y — /o — 1. But the subregional element | Pf^ is only
of order p — p. Hence | Pf, must be contained in the plane | x. This is the
signification of the supplementary equation.
13—2
196 REGRESSIVE MULTIPLICATION. [CHAP. H.
(3) The supplementary equation can be regarded as the original equation
and written in the form
where X„_i is a planar element, and P^ is a subregional element of the pth
order. The preceding proof shows that this equation is the condition that
the plane X,,_i contains the subregion Pp,
The supplementary equation is now |X„_i |Pp = 0, and signifies that the
point \X^i li^s in the region | Pp of i/ — p — 1 dimensions.
(4) The theory of duality also applies, and a?Pp = 0 can be regarded as
the condition that the subregion Pf^ contains the given point x; and the
equation, X„_iPp = 0, as the condition that the subregion P^ is contained in
the given plane X^_i.
(5) In the previous subsection it has been assumed that P^ is a regional
element, that is to say, is simple. Now let Sp be a compound extensive
magnitude of the pth order. Then in general it is impossible to satisfy the
equation a?jSip = 0, except by the assumption that a? = 0.
For aSp is an extensive magnitude of the p + 1th order ; but this manifold
v\
is defined by j—— — — ;j ^^— ^^^ independent units (cf. § 94). Hence if
xSp = 0, the coefficient of each of these units, as it appears in the expression
ocSp, must vanish. Thus there are ; :rrr-? tt-. equations to be
(iz-p-l)! (p + 1)!
satisfied. But in ic(=2fe) there are only i;— 1 unknowns, namely, the ratios
of fi, ^2 ... ^p- But if p be any one of the numbers 2, 3 ... i/ - 2,
(p-p-iy.(p + i)i
In these cases the requisite equations cannot be satisfied. If p = 1, then Sp
is a point and must be simple : the equation ooSp = 0 then means that x = 8p.
If p = i; — 1, then Sp is a planar element and must be simple (cf. § 97,
Prop. IV).
(6) Let P^_i be the planar element
Then tti, tt^.-.tt,, are the co-ordinates of the planar element P„_, with
respect to the reference elements e,, e^ ... «„.
Also if a? be 2f e, then
(xP^i) = (TTif , + TTaf a 4- . . . + TT^f ^) (e^e^ . . . e„).
Hence the equation {xP^i)=0, is equivalent to the usual equation of
a plane, namely.
107] APPLICATIONS AND EXAMPLES. 197
And conversely P^u ^ defined in this subsection, is a planar element in
the plane which is defined by the equation
TTif 1 + . . . + 7r„f y = 0.
(7) Another simple method of obtaining a slightly different form of a
planar element corresponding to the plane
is found by means of § 73 (2). The point in which the plane cuts the straight
line exep is by that article — , Hence by multiplying the v—X such
points which lie on the z/ — 1 such straight lines meeting in 6i, a planar
element in the plane is found to be
p=2 VtTi TTp/
Hence
n ( ^ ) = tr^e^ ... e^ — 7ra6,e5 . . . e^ + . . . + (— l)""*^ ir^eie^ . . . e„_i .
=8 \^i w-p/
p=
E
p=» \"i "p
Therefore by multiplying out the left-hand side and comparing the
coefficients of the term e^^ ... e, on the two sides,
= ir^e^z . . . e„ — TTjCies ...«„+...+(— 1)*^* ir^^e^ . . . 6„_i .
This factorization of the right-hand side of the above equation into a
product of If — 1 points forms another proof of § 97, Prop. IV.
(8) Among special applications of these theorems we may notice that
the condition that x may lie on the straight line joining a and 6 is
the condition that x may lie in the two dimensional region oho is
xahc = 0 ;
the condition that x may lie in the three dimensional region oihcd is
xabcd = 0.
(9) Let the complete manifold be of more than two dimensions so that
the multiplication of linear elements is progressive. The multiplication of
a planar and linear element together is necessarily regressive.
Then two lines ab and cd intersect if abed = 0. For this is the condition
that a, 6, c and d lie in the same subregion of two dimensions.
The point where a line ab intersects a given plane P„-i is P^-i . oh. But
by § 103 (2) (the extended rule of the middle factor)
P„-i . ab - (P^-ib) a - (P„-ia) b.
198 REGRESSIVE MULTIPUCATION. [CHAP. II. 107
If the line lie entirely in the plane, (P„~i6) = 0, and (P^io) = 0 ; hence
P„_i . oft = 0.
If the planar element be written as the product c^c^ ...c„-.i, then the
point of intersection of the line ab with it can be written CiC^ . . . c^-i . ah.
And by § 103 (2)
C1C3 . . . C„_i . ab = (C1C2 . . . C„_2fl^^) Cr-i + (— 1)" (c„-iCi . . . C-aCtfc) Cr-a + . . .
+ (- 1)" (CjA • • • c^^iob) Oi,
The last form exhibits the fact that the point of intersection lies in the
plane CiCa...c„_i; while the form (P„_i6) a — (P„-.ia) 6 exhibits the fact that
the point of intersection lies on the straight line ah,
(10) Two planar elements P„_j and Q^^i must intersect in a region of
1; — 3 dimensions, or in other words the extensive magnitude P„_, . Q„_i is a
regional element of the (v— 2)th order. Let such subregions be called sub-
planes. The magnitudes denoted by \P,-i + fiQ^^i for varying values of the
ratio X/ft are planes containing the subplane P„_i.Q„_i, common to P^i
and Qp-i.
In regions of three dimensions straight lines and subplanes are identical.
(11) If four given planes P„_i, Q„-i, R^-^ fi^^-i contain a common sub-
plane i^-aj then the four points of intersection of any straight line with
these planes foim a range with a given anharmonic ratio.
For let P^— 1 = Ly—i a, Q^—i = J^k-s ^> -^k— 1 = L^^u^ c, /S^— 1 = Ij,—2 d.
Let pq be any line, and assume that p lies in P,,_i and q in Q„_i.
Then L^^p^^vrPy-i, and L^^^q^^pQ^^i, Also let pq intersect iJ^«i and
/S„_i in r and 8.
Then r = iJ^_i . pq = (i2^_, ?) |> - (i2,,_i p) q
= - (L^-^qc) p + (Ly^pc) q = -p (Qr-i c)/? + «r (P^, c) 9.
Similarly 8 = — p (Q„_i d)p + «r (P^_, ci) q.
Hence the anharmonic ratio (pq, rs) = yX~^ ,\-. J!~^ -I .
(Q^_i d) (P.,_, c)
This ratio is the same for all lines pq; it can also be expressed as
(fl,., 6) (flf,_x a)/(flf,_, 6) (iJ^, a).
(12) If 12^., = XP^_i + /iiQ^_i, and /8f^_, = VP^_, + /i'0^_i, then c = Xa -h /tfi,
and d = X'a -h /a'6.
Also since (aP„_i) = 0 = (6Qr-i), we have
^^*' ^"V(a^,a).M(P.-i6) X>-
We also notice that | P^_i, | Q^,, | iJ^_i, and | /Sf„_, are four collinear points
with the same anharmonic ratio, X/i'/XV, as the four planes.
Note. In developing the theory of Regressive Multiplication the Au8dehnung$lehre
von 1862 has been closely adhered to.
CHAPTER III.
Supplements.
108. Supplementary Regions. (1) The supplement of a regional
element Pf, of the pth order is a regional element | Pf, of the (i; — p)th order
[cf. § 65 (4) and § 99]. The two subregions Pp and | Pf, are called supple-
mentary. In particular | x is the supplementary plane of the point x, and x
the supplementary point of the plane | x.
(2) If Pp be expressed as the product of p points jpi, jpa, ... j?p, then
taking the supplement
Hence if Pp be the containing region of the p independent points, then
Pf, is the common region of the p supplementary planes of those points.
(3) If Pp and P^ be two regional elements both of the pth order, then
(Pp I P^ is merely numerical.
Hence (P,|P;) = i(p,|p;) = (|Pp||p;) = (- iy<'-<')(|P,.p;) = (p/ip,).
(4) Thus if y lies in the supplementary plane of a?, then (y |^) = 0 ={x \y).
Hence x lies in the supplementary plane of y,
(5) Definition. Points which lie each in the supplementary plane of
the other will be called mutually normal points.
If the points a?(= 2fe) and y (= Si;^) be mutually normal, then
(6) A point x does not in general lie in its own supplementary plane,
unless it lies on the quadric
Let points which lie in their own supplementary planes be called self-
normal ; and let the quadric which is the locus of such points be called the
self-normal quadric.
109. Normal systems of points. (1) All the points normal to a given
point Xi lie in the plane | x^. Let x^ be any such point, and let a:^ lie in the
subregion \x^\x2, and x^^ in the subregion |a;i|^|d^; and so on; and finally
let x^ be the point |a?i |ai| ... fa?„-.i. Then assuming that none of these points
200 SUPPLEMENTS. [CHAP. III.
are self-normal, we have deduced a system of i/ independent mutually normal
points, starting with any arbitrary point a^.
(2) Definition, Let a system of v independent mutually normal elements
be called a normal system.
(f3) The intensities of the normal system of points as denoted by
a?!, a?3 ... a?„ are arbitrary.
Definition. Let any point p be said to be denoted at its normal intensity
when {p\p) = ^^ Note that the normal intensity of a point is not neces-
sarily its unit intensity.
(4) Then if Xi, x^, ,.. x^ be a normal system of elements at their normal
intensities, the following equations are satisfied
(^1 l^'i) = (^2 1^2) = etc. = 1, and (iCp \xj) = 0, where p^a.
If Xp = fip6i + f^^a + . . . ^^ffivy these equations can be written
f ip + ?2|p + . • . + f i^p = 1,
with j; — 1 other similar equations,
and f jpf ifl^ + f spf 2«r + . . . 4- f ,.pf w = 0,
with Ji/(i/ — 1)— 1 other similar equations.
(5) Also by § 97, Prop. I., the following equations hold
I X-^ ^— Jk-jX^X^ ... X^ , I wCj — - ^'S^i^s • • • X^ , ■ • • ) I X^ ^— /\t^XjX2 • • • **^¥ •
Hence (xi | a?,) = X^ (x^x^ . . . x^) = 1.
Therefore \ = . = — \,=:\3 =...=(— l)»^iX^.
yx^x^ ... Xy}
Also since (xj \xi) is merely numerical, then by § 99 (5)
(a?i \xi)=\(Xi \a!j)={\xi\\xi) = . — xaC^^a^s ••. ^.^ |a^A ... a;,,).
yX^Xi^ ... x^y
Hence by § 105 and by the previous subsection of the present article
(a?i|a?i) =
I wvitZ^a . • • Xpi
KXypC^ ... Xfff
\Xy I a?2^, (a?„ I iCg^, . . . yXp I ir„^
Hence (afja^a . . . x^y = 1, and therefore {x^x^ . . . a?„) = + 1.
Now if a?p be at its normal intensity, then (cf. § 89) — a?p is also at its
normal intensity. Hence by properly choosing the signs of a?!, ajj, ... a?r, we
can make {x^x^,.. x^) = 1. Thus finally with this convention
I X\ ^^ X^pC^ ... Xp , •Cj ^^ """ X^X^ • • • »C|» , * • • Xp ^ Y^ X ^ X\X^ ... tl/if _i •
(6) Hence the operator | bears the same relation to the normal system
Xi^x^,..XpdX normal intensities as it does to the original reference-elements.
Accordingly in the operation of taking the supplement the original reference-
elements may be replaced by any normal system at normal intensities.
IIOJ EXTENSION OF THE DEFINITION. 201
110. Extension of the definition of Supplements. (1) This possi-
bility of replacing the original reference-elements by other elements in the
operation of taking the supplement suggests an extended* conception of the
operation.
In the original definition the terms d, 69... 6^ represent the reference-
elements at their normal intensities as well as at their unit intensities. But
suppose now, as a new definition which is allowable by § 109 (3) and (6),
that the normal intensities of these reference-elements are €1, fj ... €„. Then
by hypothesis [c£ § 109 (3)]
and so on.
Also it must be assumed that ejCa . . . e„ (ej^a . . . e„) = 1.
1
Let €i€q . . . 6„ = a =
]««} . • • Vy
V^i^s • • • ^v)
Then ki = ~i ^»^ . . . 6„, 1^3 = — ^ ^1^8 • • • ^v, and so on.
€1 €.2
(2) This extended definition in no way alters the fundamental properties
of the operation denoted by | . For this operation has been proved to be referred
to an indefinite number of normal sets of points and cannot therefore be
dependent on the symbolism by which we choose to denote one set of them.
Thus it follows that the symbol | obeys the distributive law both for
multiplication and addition. Also ||Pp = (- l)'*^*'"'*^Pp, where Pp is of the
pth order.
But (ey\ei)=-, («2ka)=-5^,...(«. kr)= -3.
€1 Cj €y
Also it is not necessary that 61, €, ... €y should all be real; thus any number
of their squares may be conceived as being negative.
(3) The self-normal quadric is defined by the equation (a? |a?) = 0;
that is by ^i^ei^ + f »V«a» + . . . f .V^r' = 0.
If 61, €, ... 6^ be all real, this quadric is purely imaginary: but if some of
them be pure imaginaries, this quadric is real. Since only the ratios of
€i,€2...€y are required for defining the self-normal quadric, it is allowable
when convenient to define, €i€, ... e, = 1. Hence in this case (^le, ... e^) = 1.
(4) The equation of the supplementary plane of any point x is (y \x) = 0;
that is, if XT = 1(e and y = Si/^, the equation
f il/i/ci* + f aW^a' + . . . + f ..l/KAr' = 0.
But this is the equation of the polar plane of x [cf. § 78 (1)].
Hence the method of supplements is simply a symbolic application of
the theory of reciprocal polars and its extension to linear elements and to
other regional elements in manifolds of more than three dimensions.
* This eztenaion is not given by Qrasamann.
202 SUPPLEMENTS. [CHAP. III.
(5) Normal sets of elements are obviously sets of polar reciprocal
elements forming a self-conjugate set with respect to the self-normal quadric.
In future it will be better to speak of taking the supplement with
respect to an assumed self-normal quadric, rather than with respect to a
particular set of normal elements.
111. Different Kinds of Supplements. (1) It may be desirable to
take supplements with respect to various quadrics. The operation of taking
the supplement with respect to one quadric is different jfrom the operation
of taking it with respect to another. If one operation be denoted by the
symbol |, let another be denoted by the symbol I. Then \P and IP denote
different extensive magnitudes. But the operator I possesses all the proper-
ties which have been proved to belong to the operator |.
Also if the supplement is taken with respect to a third quadric, the
operator might be denoted by Ii and so on.
(2) Confining ourselves to two operations of taking the supplement,
denoted by | and I, we see that the two self-normal quadrics are denoted by
(x\x) = 0, and (a?la?) = 0.
But [cf § 83 (6)] in general two quadrics possess one and only one system
of p distinct self-conjugate points.
Let 6i , e^ . . . 6„ be these points and let €i, €a . . . 6,, be their normal intensities
with respect to the operation |, and e/, €a'... e/ those with respect to the
operation I.
Then CiC^ . . . e^ = ^ x = €,€2' . . . e/ = A.
Hence («,|^)= ^ ^g + g+ ... + g),
and (^l^) = A(|; + g,+ ...+g,).
Also ki = — i^8"* ^rj and Iei^—r„e^z'"^^'
€1 €1 -■
112. Normal Points and Straight Lines. (1) The following propo-
sitions can easily be seen to be true for mutually normal points with respect
to any quadric.
On any straight line one point and only one point can be found normal to
a given point, unless every point on the line is normal to the given point.
If a be the given point and be the given line, this point is
6c |a = (6 |a)c — (c |a)6,
unless be |a = 0 = (a \b) = (a |c).
Ill — 113] NORBiAL POINTS AND STRAIGHT LINES. 203
(2) There are two exceptional self-normal points on every straight
line (viz. the points in which the line cuts the self-normal quadric), but in
general these self-normal points are normal to no other points on the line.
If however these two self-normal points coincide, so that the line is tan-
gential, then this double point is normal to every other point on the Una
(3) It follows from the harmonic properties of poles and polars that
the pairs of normal points on a line form a system of points in involution,
with the self-normal points as foci.
(4) This harmonic theorem can be proved thus: let ai, a, be the two
self-normal points of any line ; then (oi jO]) = 0 == (a, la,).
Let Xoi + fia^ and X'ai + /Lt'o, be any pair of normal points. Then
Hence (X./i' + X'/i)(ai |a,) = 0.
Hence X//a = — X'//*'.
113. Mutually normal regions. (1) Two regions Pf, and P„, where
p and a denote the orders of Pf, and P^, respectively, are called mutually
normal, or normal to each other, if every pair of points p^ and p^ respectively
in Pf, and P^ are mutually normal.
(2) Let Pp be defined by the points /)Jf\ p^*\ "-P^\ ^^^ -P# by the points
P^)\pT> "'P^J^' Then any point on P^ must lie on the intersection of the
supplementary planes of p^\ pf\ . . . pl^\ Similarly any point on P^, must
lie on the intersection of the supplementary planes ofjp^J), p^^\..,p^^' Thus
the condition thatP, and Pp should be mutually normal is that P^, should
be contained in | P^y or that Pp should be contained in |P«y. Either condition
is sufficient to secure the satisfaction of the other.
(3) If Pp and P^ be mutually normal, then
a ^ p — p, that is v ^ p + a.
(4) If p = p + a, then P^ = | Pp. Hence the supplementary regions are
mutually normal. The supplementary region of Pp will be called the
complete normal region of Pp, or (where there is no risk of mistake) the
normal region of Pp . Thus the supplementary plane of a point is its normal
region.
(5) In any subregion Pp (of the pth order) p mutually normal points
can be found, of which any assumed point in Pp (which is not self-normal)
is one. For [cf. § 78 (9)] take a?j to be any point in Pp, then | Xi intersects
Pp in a region of the (p — l)th order. Take ^ to be any point (not self-
normal) in this region. Then | .r, intersects this region in a subregion of the
(p — 2)th order ; take a?, (not self-normal) in this subregion of the (p — 2)th
order, and so on.
204 SUPPLEMENTS. [CHAP. IIL
If however Pf, lie in the tangent plane to the self-normal quadric at one
of the self-normal points lying in Pf^y then this self-normal point must be one
of any set of p mutually normal points in Pf^. For the supplementary plane
of the self-normal point by hypothesis contains Pf,, hence the supplementary
plane of any point in Pf, contains the self-normal point. Thus proceeding
as above in the choice of a?j, a?a, etc., the last point, a?p, chosen must be the
self-normal point.
(6) To find the subregion (if any) of the highest order normal to P^
which is necessarily contained in P^ ; where p and a are respectively the
orders of P^ and P^.
Any region normal to P^, is contained in |Pp. Now |Pp and P^ do
not necessarily intersect unless (i/ — p) + <r > y, that is, unless a> p.
Assume <t > p. Then [cf. § 65 (5)] |Pp necessarily overlaps P^ in a
subregion of the {(i/-/o) + o- — i/jth order, that is, of the (o- — p)th order.
Every point in this subregion is necessarily normal to Pp\ and hence this
subregion of the (<7 — p)th order, contained in P^, is normal to Pf,, If the
intersection of P„ and |Pp is not of a higher order than {a — p), the regional
element P^ |Pp defines it ; thus if P,, |Pp be not zero, it is the subregion of P^,
normal to Pf,,
(7) By subsection (5) p mutually normal points can be found in P^
and {a — p) mutually normal points can be found in the intersection of P^
and I Pp {a>p). Also by the previous subsection each point of the one
set is normal to each point of the other set. Thus the a points form a
mutually normal set. Hence it is easy to see that, given two subregions
Pa and Pf, (a > p), <r mutually normal points can be found in them, and of
these p (or any less number) can be chosen in P^ and the remainder in
Pff ; also that any one point in P^, can be chosen arbitrarily to be one of
these points or (if p points are to be taken in Pf^) any one point in the inter-
section of P«r and I Pp.
114. Self-normal Elements. (1) Every element in a subregion
defined by p independent self-normal elements mutually normal to each other
is itself self-normaL
For if Oi, di, ... ap be such elements,
(oi loi) = 0 = (oa loj) = eta = (oi K) = etc.
Hence {(XjOi + XjO,-!- ... +\ffip)\{\ai + \(h + ... + ^p«p)1 =0.
Also any two elements of such a subregion are normal to each other.
For {(XiOi +X^+ ... -hXpttp) KmiOi + /AaO, + ... +fif/ip)} = 0.
(2) Accordingly such a subregion is itself a complete generating region
[cf. § 79] of the quadric, (a? |a;) = 0 ; or is contained in one.
\
114] SELF-NORMAL ELEMENTS. 205
But from § 79 the generating regions of this quadric are, in general,
of v/2 — 1 or (y — 1)/2 — 1 dimensions according as i/ is even or odd.
Hence sets of v/i or (i/— 1)/2 (as v is even or odd) self- normal and
mutually normal elements can be found.
(3) Also by § 80 a set of conjugate co-ordinates Ji, J2 .•• , ^i> ^j ••. can
be found all self-normal and all mutually normal except in pairs, i,e. {ji \ki) is
not zero, nor ( j, | Atj) and so on. But {ji \ji) = (^'1 l^'a) = etc. = (ji \k^) = etc. = 0.
If V be even, v such co-ordinates can be found which define the complete
manifold ; but if v be odd, v—1 can be found, and one co-ordinate element e
remains over, which can be assumed to be normal to the v — 1 other elements,
but not self-normal.
(4) Firstly let v be even. Let 61, ^a, ... 6^ be a set of normal elements,
€1, €2'»' €^ being their normal intensities according to the notation of § 110.
Then by § 80, we may assume
ji = \ (€161 + ie^), ki = X, (€161 - %€^),
ja = ^ (€^ + i€4«4), h = \2 (^s^s - ie^e^X
Jp — A»^ \€p^i6p—\ -r ^^r^r/i f^w — ^v K^v—i^v—i — t€p6p).
2 2 2 2
Hence jjci = — ZiXi^e^ie^y with « — 1 other similar equations.
Thus (jikJA "J^h) = {- 2if VXa^ . . . X^,
2 2 2
Again | j, = Xi (€1 1 e, + ie^ | ^2) = — (^262^5 . . . e„ — ieyeie^ . . . e^).
^1^2
.--1
But jjc^jih ' ' • ir^F = (- 2i)2 €564 .. . e^Xj'X,^ . . . \^e^^ . . . 6„ ;
2 8 2
hence jij^k^jjc^ . . . jjc^ = (— 2if — XjXa'X,* . . . X^' {^le^e^ . . . c,, -h ie^^ ... 6^)
2 2 ^1^2 2
2 2 2
(5) Now let X,, X9, ... X„ be so chosen that
2
Xi = Xj = . . . = X„ = -7^ .
1 1
Then j, = -^ (6,^5 - »e,e,), *' = 7^ (^»^ "*" *^'^')'
1 . 1 .
is = -^ (€4^4 - ^€363), ** "^ 7/2 ^^*^* ^ ^^^'
1 . 1 .
2 V^ 2 V-^
206 SUPPLEMENTS. [CHAP. III.
And (ji^ijaA:, ...>A?;) = (-2»)»f-2-j =1*.
2 2
Hence jij^k^jzh • • -jV^f = *' lii- Similarly jajjAri^a Atj. . .jVA:„ =i* | ja, and so on.
2 2 2 2
When \i, Xj, ... X„ have been chosen as above, the conjugate self-normal
2
elements will be said to be in their standard normal form.
When the self-normal elements are in this form
¥
jA = te,€2e,ft2 ; (ii I ii) = (*^i I ji) = * * (kijijih . "jvk^) = - 1.
2 2
(6) Secondly, let v be odd, and let «, Ci, eg, ... e„_i be the set of normal
elements with normal intensities €, 6i, ... 6^i.
Let the standard normal forms of the conjugate self-normal elements be
j^ = -^ {€^ - i€iCi), ^'^J2 ^^^ "*" *^'^^^*
1 . 1 .
1 . 1 .
Thenj,A;i = i€i€jeiCa, with similar equations.
xience v^i'^i^a"^ • • * ^^ — i**'!' — i' ^'^ * ^1^2 • • • ^i^— 1 Ky^i • • • ^v—\) * ^ •
~2" "2~
Also ejij^k^ ...>-iA?F-i = — i * €~*lii, with similar equations for the other
"2" ~2~
elements.
And (j,|A;0 = (A,|iO = -l.
116. Self-normal Planes. (1) Let a be a self-normal element ; now
the region \a contains all the self-normal elements which are normal to a.
Hence \a contains all the generating regions of the quadric which contain a.
Therefore | a is the tangent plane to the quadric at a.
(2) The plane-equation of the quadric is, (X|X)=0, where X is any
planar element. For this equation is the condition that the region X
contains its supplementary element {X.
A tangent plane X, for which (X\X)=0, will be called a self-normal
plane.
116, Complete Region of Three Dimensions. (1) The application
of these formulae to a manifold of three dimensions is important. Consider a
116:— 117] COMPLETE REGION OF THREE DIMENSIONS. 207
skew quadrilateral jij^kik^ formed by generators of the self-normal quadric ;
so that jiji and kik^ are two generators of one system, and j^k^ and j^ki are
two generators of the other system.
A self-conjugate tetrahedron 616^3^4 can be found such that if ^ be the
point ^^e, the self-normal quadric is
the normal intensities of ^i, 6,, ^, 64 are then 61, €2 > €3, 64, and
(W3e.) = A = ^-^^^.
1 1
(2) Assume j, = -^ (e^ - t€,ei), *i = "^ (^^ + ^^i^O*
1 . 1 .
Hence (jikij^kq) = i* = — 1.
Also ji ja*2 = - I ji, jaiA = - 1 ja, Aifc^jj = - | A:,, kikjj^ = - 1 A:a.
Thus I j, J2 = j, jjAa . J2Jiki = - Ui^ijMJd2 ^jiia ;
and similarly | kik^ = ^li^.
Also ljA = -jA, Ijiki^-j^ki.
(3) Hence for a generator (0) of one system of the self-normal quadric
\0 = 0, and for a generator G' of the other system | G' = — 0', Let the
system of generators to which 0 belongs be called the positive system, and
that to which 0' belongs be called the negative system.
117. Inner Multiplication. (1) The product of one extensive mag-
nitude (such as Pp) into the supplement of another extensive magnitude
(such as I P^) is of frequent occurrence ; and the rules for its transformation
deserve study. These rules are of course merely a special application of the
general rules of progressive and regressive multiplication, which have been
explained above.
(2) This product Pp|P# may also be regarded from another point of view.
Since P^ \ (P, -h P/) = Pp | P^ + Pp | P/, we may conceive [cf. § 19] the symbol |
not as an operation on P^ but as the mark of a special sort of multiplication
between Pp and P^,. Let this species of multiplication be called * Inner
Multiplication,' and let the product Pp|P^ be termed the inner product of P^
and P^. In distinction to Inner Multiplication Progressive and Regressive
Multiplication are called Outer Multiplication.
(3) It is obvious that inner products and inner multiplication must be
understood to refer to a definitely assumed self-normal quadric ; and further
that, corresponding to different self-normal quadrics, there can be different
sorts of inner multiplication. But general formulae for the transformation of
such products can be laid down.
208 SUPPLEUEMTS. [CHAP. III.
118. Elehentabt Transformations. (1) Let P^ and P, be extensive
magnitudes, simple or complex, of the pth and crth orders respectively.
Then i» — <r is the order of |P». The product P^ \P, is progressive if
p + (p — a) < V, that is, if p < «r ; and is regressive, it p><r.
(2) Ifp<ai P,\P, = (-iyi'-')\P,.P,-
and hence |(Pp |P,) =(- 1)p('-') ||P, . \P,.
But by § 99 (7), \\P. = (- !)'<'-'' P,.
Therefore finally, | (P, \ P,) = (- 1 )<"+") <- ') P, | P,.
(3) If pxr; then|(Pp|P,) = |Pp.||P,=(-l)'(-')(|P,.P,)
= (_ i)»(r-.r)(_ i)»(K-/.)(p^ I p^) = (_ l)'(P+»)(P, I p^).
(4) If p = <r ; then (Pp | P,) is merely numerical : write Pp' instead of P».
Then (Pp|p;)=|(P,|p;)=(p;|Pp).
119. Rule of the Middle Factor. (I) The extended rule of the
middle factor can be applied to transform P^ | Q^, where P^ and Q^ are simple
magnitudes of the pth and o-th orders respectively. In the first place
assume that p > o*. Let the multiplicative combinations of the o-th order
formed out of the factors of the first order of P^ be P"^ P®, etc., and let
P, = P^^P^L^ = P?P?.^ = etc.
Then from the extended rule of the middle factor, we deduce
i'p|Q. = (n"|Q.)P;i. + (P?|Oa)Pp«-. + etc (1).
(2) Secondly, assume that p<a;
Then (Pp I Q,) = (- 1)(p+')(-p) | (Q, | P,).
Now let Q^\ Q^, etc. be the multiplicative combinations of the pth order
formed out of the factors of the first order of Qa, and let
Q.=Q!IU% = <2?Q?-p=etc.
Then by equation (1) of the first case
(Q. I P,) = (Q?' I P,) QS-, + (Q? I -Pp) <2?-p + etc.
= {P, I OS!) (2?-p + (^p I <3?) <2?-p + etc.
Hence (- 1)(p+')(-p) (P^ | Q,) = (P^ | q») | Qo.,^ + (P^ | Q») | (2«_^ + etc. . . .(2).
The formulae of equations (1) and (2) will be called the rule of the middle
factor for inner multiplication.
120. Important Formuijl. (1) The rule of the middle factor does not
apply when both factors are of the same order. But the transformation in
this case is given by § 105. For if each factor be of the pth order, then
(OiOa . . . ap 1 6162 . . . &p) = (oiO, . . . ap . I &i I &2 • • • | ^p)
(aa|6,), (Oalftg), («2|M
(ap |6i), (dplft.) (ap|6p)
.(3).
118—122]
IMPORTANT FORMULA.
209
Important special cases of this formula are
{didi ... Op I OiO^ • • • CLp) =
(ttilcti), (oilo,), (Oilttp)
(OaK), (O^lo,), (Oalttp)
(aploi), (ap|a,), (apl^p)
(oittj 1 6,62) = (a, I 6,) (a, 1 62) - («! 1 6s) ((h 1 61) ;
(a,«j I OiO,) = (a, I aO(as | a^) - (a, | a,)».
(2) Also if the complete region be of r - 1 dimensions, the products
(aitt, . . . a^) and (6162 • . . K\ although merely numerical, may each be conceived
as progressive products. The proof of § 105 still holds in this case, and
therefore
(oia, . . . a„) (6162 . . . 6„) = {oiOi . . . a„ 1 6162 • • • b,)
(a, |6i), (a, I62), (a, |6^)
(a, |6i). (oal^a), ((hlK)
* •
(a^ |6i)» (ar|6j), (a^lM
This is the ordinary rule for the multiplication of two determinants.
121. Inner Multiplication of Normal Regions. If il, 5, C be
three mutually normal regions, (so that the multiplication ABC must be
pure progressive), then
(ABC\ABC)==(A\A)(B\B)(C\C) = {AB\AB)(C\Cy
This theorem can easily be proved independently; but we will deduce
it at once from the formula for (aiO, ... a^ 1 0102 ... a^) of § 120.
For let A = OjO, ... a^, B = 616a ••• K, C= CiCa-.c, ; then each of the groups
(tti, Oa ... ap), (61, 6a ••• 6a), (Ci, C2 ... Cr)
may be conceived [cf. § 113 (5)] to consist of mutually normal elements. But
since A, B, G are mutually normal regions, it follows that the whole set of
p + <r + T elements are mutually normal.
Hence
{ABC I ABC) = (o^iOa . . . ajbib^ . . . 6,CiCa . . . c^ | a, . . . 6i . . . Ci . . . Ct)
= (Oi I ChXOa I Og) ... (6, I 61) ... {K I K) (Ci I C,) ... (Cr I C^).
Also {A\A) = (a,\a,){cu, \a,) ...(a,\a,l and (J9|5), (0|(7) and (AB\AB)
are equal to similar expressions. Hence the theorem follows.
122. General Formula for Inner Multiplication. (1) Equations
(1) and (2) of §119 can be extended so as to prove two more general
formulse which include both them and equation (3) of § 120.
Consider the product -Pp+»|QpQr, where P,i+9, Q,,, Qr represent simple
magnitudes of the {p + o')th, pth, and rth orders respectively.
In the first place assume that a- >r.
w. 14
210 SUPPLEMENTS. [CHAP. III.
Then P^\Q,Qr='P^(mQr).
But since <r > t, the product is a pure regressive product and is therefore
associative. Hence
Now let P^\ P^, etc. be the multiplicative combinations of the pth order
formed out of the factors of the first order of Pp+», and let Pa\ P?, etc. be
the multiplicative combinations of the <rth order formed out of the factors
of Pp+», so that
P,+, = P^^PS^ = P^P? = etc.
Hence by equation (1), P,+, | Q, = 2 (P^^^ | Q,) P^^\
Therefore finally P^,\. Q,Q, = 2(pW|(2,)P:*> |Q, (4).
(2) Secondly, let <r<T. Let QpQt = Qp+t, and let Q^\ Q^, etc. be the
multiplicative combinations of the pth order formed out of the factors of the
first order of Qp+n and let Q?, Q? be those of the rth order, so that
Also let Pp+^r^PfiP^, where P^ is of the pth order, and P^ of the <rth
order.
Then P^ I Q,^ = (- 1) ('+-)(-^-') \(Q,^\P,^).
But by equation (4) Q^, \P,P, = t (Q<*> |P,) Qi^^ \P,.
Hence
KQ^r \P,P,) = 2(<3^*' |Pp) \(Qi'^ IP.) = ( - 1)('+^)<'-') 2(P, \Ql'^)P, \Qi'\
A A
Therefore finally, P.P. \ Q^, = (- 1)" ('+^) S (P, | Q^*') P. | Qi*> (5).
A
(3) Equations (4) and (5) are more general than equations (1), (2) and
(3) but the readiness with which the equations first found can be recovered
from the extended rule of the middle factor makes them to be of the greater
utility.
The theory of Inner Multiplication and the above formulae are given in
Grassmann's Ausdehnungslehre von 1862.
123. QUADRICS. (1) The theory of quadrics can be investigated by
the aid of this notation. Let the quadric be chosen as the self-normal
quadric for the operation |. Then the equation of the quadric is {x |^)= 0.
Let the reference points «i, e^..,e^ be any v independent elements, not
necessarily mutually normal. Then if a? = 2f e, the equation of the quadric
according to the notation of Book III., Chapter III. is written
{al^xf = a„fi« + ... + 2awfif,+ ... = 0.
But {x\x) = {e, k)fi'+... + 2(ci |eB)fifa+....
Hence we may write, {e^ \e^ = 0,1, (Cj leg) = Oa, etc., (e, |«a) = »i2, etc.
i
123] QUADRICS. 211
(2) Since by § 120 (e^ei . . . e^y = (eie^ . . . ^^ |e,ea . . . e^)
(«a|«l)» («8|«8)i---(«2|Cr)
it follows that (eiea...^^)* is the discriminant of the quadratic expression
(w \x).
Since (ci«a ... «r) cannot vanish («i, 6^, ... e„ being independent), it follows
that the quadric cannot be conical.
(3) The equation of the polar plane of any point x becomes {x \y) = 0.
The plane-equation of the quadric is (X„_i |Jr„_,) = 0; where X^i is a
planar element.
The equation of the polar point of any plane X^,, becomes
(4) Let bx be any line drawn through a given point 6; and let this
line intersect the quadric in the point X6 + ^.
Then X*(6 \b)+2\fi(b \x)-^fi^(x \x)^0 (1).
This quadratic for \/fi in general gives two points on the quadric.
(5) These points coincide if (6 \b)(x\x)— {b [«)" = 0.
This is the equation of the tangent quadric cone with vertex 6.
But (6 \b) (x \x) - (b \xy = (bx \bx).
Hence this cone can be written (bx \bx) = 0.
(6) The identity (b \b)(x \x) - (b |a?)» = (6a; \bx\ can be written
(aJiby (alUxy - [{aJibJix)Y = 2 {fi^. - fi.^,) (fiK^l. - )8^f x) (e^. \e,e,).
Also (efie^ \exe^) = (e^ \ex) (e^ M - (e^ l«^) (e^ le^) = a^xo^r^ - oLpi^oi^k-
(7) The roots of the quadratic equation (1) are
X. -jb \w) + V{(6 \<ry - (x \x) (b \b)}
/*! " (b \b)
X, -(b\a>)-^{(b\xy-(x\a!)(b\b)]
^~ (f>\b)
(8) If ai and a, are the points Ti^b + fiiX, and Xjb + fi^, then the an-
harmonic ratio (o^o,, 0:6) is
-(6 |a;)- V|(6 k)» - (^k)(6 |6)J ^ (6 k) + VK6a; |6a?)}
- (6 k) + \/l(6 k)» -(a? k)'(6 16)} (6 \x) - V{- (6^ \bx)] '
(9) Firstly let (6 |a?)» < (a? |aj) (6 |6).
(6 IxY
Let 0 be such that cos" 0 = . , wr il\ 5 ^"^<5[ 1®* P = (^^» ^)-
(a?|a?)(6|6)
^-^ cos d + 1 sm d ^^
Then p = ^ — ^-^—a-^-
'^ cos ^ — t sm ^
14—2
212 SUPPLEMENTS. [CHAP. III.
Also sin* 6 — . , . /tttt •
{x \x){h \b)
Therefore we deduce the group of equations
^ = ^l<>gP=COS "77/ ■ V /L -T'L\^ =Sin ^a/ (/ I v/L .Lxt ••••(2).
(10) Secondly, let (6 \xy > {x \x){b \h\
Put cosh* e = ,-^-^^rT .
{x \x)(b \b)
Then ^ cosh 6 + sinh g ^ ,^
'^ cosh 5 — sinh 0
Also 8mh'g= -<^'^> ■
(x \x){b \b)
Hence we deduce the group of equations
^ = s lo}? P = cosh~* —r, — . [ ;. — TTi = sinh""* * / ri — , \ /l ,lvi • • • (3)«
2 ^'^ VK^ k)(6 ;6)1 V K^ l^)(* I*)}
(11) If (6a? \bx) be positive for every pair of elements 6 and x^ it follows
from § 82 (4) that the quadric, {x\x) = 0, is imaginary.
If the quadric be a closed real quadric and b and x both lie within it, or if
both lie without it and the line bx cut the quadric in real points, then it
follows from the same article that (bx \bx) is necessarily negative.
124. Plane-equation of a Quadric. (1) Taking the supplement of
the equation, (6a; | 6a?) = 0, and writing B instead of ■ 6 and X instead of | x,
we find the equation {BX\ BX) = 0, which can also be written
(B\B)(X\X)^(B\Xy = 0,
This equation [cf. § 84 (4)] Ls the plane equation of the degenerate
quadric enveloped by sub-planes lying in the plane B and touching the
quadric.
(2) Again, by a proof similar to that in § 123, let B and X be any two
planes, and let the two planes through their intersection BX which touch
the quadric be A^ and A^. Also let p be the anharmonic ratio of the range
[BX, AiAq\,
Then if (B\Xy<(B\B)iX\X),
._1, _ (B\X) ■/ (BX\BX)
"-ai'ogP-^os ^{(X\X)(B\B)}-^''' V {(X\X){B\B)}-
And if (B\Xy>(B\B)(X\X),
„ 1, t._, (B\X) .., /-{BX\BX)
(3) Again, let Z^ be any subregion of p — 1 dimensions which touches
the quadric. This condition requires that Zp should lie in the tangent plane
124] PLANE-EQUATION OF A QUADRIC. 213
to the quadric at some point 6, and should contain b. We can prove that
the condition to be satisfied by £p is, (Lp \ L^ = 0.
For let ^1, {,,... {p, be assumed to be p mutually normal points on Zp,
which is possible according to § 113 (5).
Then by § 120 (1) {L,\L,)^(l,\k){k\U) ...(/,|Zp).
Hence if (Zp | Zp) = 0, then one at least of the points Zi, Zs, ... Zp must be
self-normal. Assume that Zp is the self-normal point 6. Then the remaining
points Zi, 2a, ... Zp_i all lie on the plane | 6, which is the tangent plane of 6.
Thus Pluckers conception of the line equation of a quadric in three dimen-
sions can be generalized for any subregion in any number of dimensions*.
(4) Consider the four subregions 5p, Xp, udp, A^, of p — 1 dimensions
which lie in the same subregion of p dimensions. Then considering this
containing subregion as a complete region we see that 5p, Xp, -4p, A^ have
the properties of planes in this region.
Let ilp and Af! both contain the subregion of p — 2 dimensions in which
J9p and Xf, intersect. So that Af, = XjBp + fiXp, and A/ = X'B^ + A^'Xp. Then
the four subregions J9p, Xp, Ap, Ap form a range with a definite anharmonic
ratio XpljX'fjL] let this ratio be called p. Let Ap and Ap touch the self-
normal quadric.
Then Xjp, and X'jp! are the roots of the quadratic
\-{Bp\Bp)^2\p^{Bp\Xp) + p?{Xp\Xp)^0.
Hence, as before, if {Bp \ Xp^ < (Bp \ Bp) {Xp \ Xp),
then 5 = ^.logp=cos-' (^/»I^p)
2r-«^ -" ^[(Bp\Bp)(Xp\Xp)}'
And if (Bp\Xpy>(Bp\Bp)(Xp\Xp),
1 / D I Y \
then ^ = 2 '"« '^ = *^*^"' vmri) (ip i"^)i •
It is to be noticed that the formulae for sin 0 and sinh 0 do not hold unless
p be unity or v — 1.
* As far as I am aware this generalized form of Pluoker*8 line-eqnation has not been given
before.
CHAPTER IV.
Descriptive Geometry.
125. Application to Descriptive Geometry. An extensive manifold
of y — 1 dimensions is a positional manifold of i/ — 1 dimensions with
other properties superadded. These further properties have in general no
meaning for a positional manifold merely as such. But yet it is often
possible conveniently to prove properties of all positional manifolds by
reasoning which introduces the special extensive properties of extensive
manifolds. This is due to the fact that the calculus of extension and some
of the properties of extensive manifolds are capable of a partial interpreta-
tion which construes them merely as directions to form * constructions ' in a
positional manifold. Ideally a construction is merely an act of fixing attention
upon a certain aggregate of elements so as to mark them out in the mind
apart from all others ; physically, it represents some operation which makes
the constructed objects evident to the senses. Now an extensive magnitude
of any order, say the pth, may be interpreted as simply representing the fact
of the construction of the subregion of /> — 1 dimensions which it represents.
This interpretation leaves unnoticed that congruent products may differ by
a numerical factor, and that, therefore, extensive magnitudes must be con-
ceived as capable of various intensities. Accordingly, in all applications of
the Calculus to Positional Manifolds by the use of this interpretation it
will be found that the congruence of products is the sole material question,
and that their intensities can be left unnoticed ; except when the products
are numerical and are the coefficients of elements of the first order which
have intensities in positional manifolds. The sign of congruence, viz. = [cf.
§ 64 (2)], rather than that of equality is adapted to this type of reasoning.
Also supplements never explicitly appear, since they answer to no mental
process connected with this type of reasoning.
126. Explanation of Procedure. (1) In the present chapter and in
the succeeding one some applications of the calculus to positional manifolds
are given. Except in § 130 on Projection, the manifolds are of two dimensions.
125 — 128] APPLICATION TO DESCRIPTIVE OEOMETRY. 215
and the investigations form an example of the application of the Calculus to
Descriptive Geometry of Two Dimensions. Other applications of this type
have already been given in §§ 106, 107.
(2) In two dimensional complete regions the only products are of two
points which produce a linear element, of two linear elements which produce
a point, and of a point and a linear element which produce a numerical
quantity. If a product yields an extensive magnitude, the act of using such
a product is equivalent to the claim to be able to construct that subregion
which the magnitude represents. Thus the product ab represents the indefi-
nitely produced line joining ab, and the use of the product is the equivalent
to drawing the line. Similarly if L and L' are two linear elements in a
plane, the use of LL' is equivalent to the claim to be able to identify the
point of intersection of the two lines L and i', which by hypothesis have
been constructed. Thus the representation of a point by a product of certain
assumed points is the construction of that point by drawing straight lines
joining the assumed points and is the point of intersection of lines thus drawn.
127. Illustration of Method. The method of reasoning in the
application of this algebra to Descriptive Geometry is exemplified by the
proof of the following theorem*.
If abc and def be two coplaoar triangles, and if « be a point such that
sd, 86, sf cut the sides be, ca, ab respectively in three colliuear points, then
sa, sb, 8C cut the sides ef,fd, de respectively in three collinear points.
For by hypothesis
(sd . be) (se . ca) (sf. ab) = 0.
Hence by the extended rule of the middle factor
{(sdc) b - (sdb) c] {(sea) e - (see) a] [(sfb) a - (sfa) b] = 0.
Multiplying out and dividing by the numerical factor (abe),
(sde) (sea) (sfb) - (sdb) (see) (sfa) = 0.
The symmetry of this condition as between the triangles abe and def
proves the proposition.
128. von Staudt's Construction. (1) Let a, e, b represent any three
points in a two-dimensional region, which are not collinear.
In ac assume any point d arbitrarily, and in eb assume any point e.
Since the intensities of a, c and b are quite arbitrary, we may assume that
d^a + Se, 6 = 6 + €C,
where S and e are any assumed numerical magnitudes.
Then it is to be proved that any point x on a>c (%.e. of the form a + fc) can
be exhibited as a product of the assumed points, or in other words can be
constructed. This construction to be given is due to von Staudtf .
♦ Due, I belieye, to H. M. Taylor. t Oeometri€ der Lage, 1S47.
216
DESCRIPTIVE GEOMETKY.
[chap. IV.
(2) Firstly, consider the following products, or in other words make the
constructions symbolized by them :
q==ae.db, qi = qc. de, pi = qj) . ac.
Thus pi = ae.db.c .de.b .ac.
Then pi is the point a + 2Sc.
Fio. 1.
For q = (ab + edc) (ab + Scb) = Sab . cb + eac . ah + eSa^) . cb
= 8 (acb) b + € (acb) a+ eS (ax)b) c = — (abc) {ea +Sb-\- eSc]
= ea + 86 + eSc ;
where the numerical factor — (abc) has been dropped for brevity. This will be
done in future without remark.
qi = (eac + Sbc) . de = {eac + hbc) {ah -^-ea^c^- hcb)
= eac . ah + heac . cb + Sbc . ah + Sebc . a^
= € (ach) a + Se {acb) c + 8 {acb) 6 + Se {a/ib) c^ea + Sb^- 2S€C.
Pi = grjj . oc = e {a + 2Sc} b,ac = € {ahc) {a + 28c} =a + 28c.
Hence j^i is the point a + 28c.
(3) Again, substitute d for a and ^, for d in the above product. The
new lines in the figure are represented by dotted lines. Then since pi=d + Sc,
we obtain the point
Pj = (fe . pib . c .piB . 6 . dc = d + 28c = a + 38c.
Similarly by substituting pi for d and j>, for pi in this construction, we find
p, = a + 48c, and so on successively. Thus finally if v be any positive integer,
we find p„ = a + (i' + 1) 8c.
J
128]
VON STAUDTS CONSTRUCTION.
217
(4) Secondly, consider the point e' = gpi . 6c = 6 — ec (cf. fig. 2).
Make the following constriction,
r^^aef .de, p^=^ rj} . <ic = cie . de , b . etc.
Now
Vi = (db — eac) (ah + edc + Scb) = edb . dc + Sab . cb — ea^ . aft — Seac . c6
= — 2€ac . oft + &i6 . cb — Scoc . c6 = — 2€ (acb) a + S (acb) 6 — Se (ac6) c ;
and Pj= — (ao6) [^eab + Secb] .a/) = a + ^Sc,
(5) Similarly by substituting p, instead of d in this construction we
obtain
Fio. 2.
and by continually proceeding in this manner we finally obtain if i^ be any
positive integer
(6) Then if /i be any other positive integer, the construction of the first
figure can be applied /a times starting with j> instead of with d. Thus the
point ^
can be constructed.
^ 2'
2"
218
DESCRIPTIVE GEOMETRT.
[chap. IV.
(7) Thirdly, make the following construction (cf. fig. 3)
q' = ab . defy d' = €[e . ac = ab . de' .e.dc.
Fio. 3.
Now q =ab[ah — eac + hcb] = — eab . ac -h iab .cb = — (abc) {ea + S6j,
d' = (eas + Bbe) etc = {eab + ^ac + Sebc) ac
= cab . ac + Sebc .ac = € (abc) {a — 8c) = a — Sc.
(8) In this construction if we substitute for d any constructed point of
the form
we obtain
p =
2*'
a +
2"
Sc,
P =
a-
'2"
8c.
2"
Thus all points of the form (a± Si,^<^] ^^^ ^^^ ^ constructed.
(9) Fourthly, let p, p' and jp" be three constructed points, and let
p = a + vrSc, p' = a + nr'Sc, p" = a + vr'^Sc.
Then p'=p + (fsr'^ isr) Sc.
129] VON STAUDt'S CONSTRUCTION. 219
Now in the first construction substitute p for a and p' for d. Then
we obtain
p/ =p + 2 («r' - w) Sc = a + (2iir' - «r) Sc.
Similarly by substituting p/ for a, and p" for d, we obtain
Pj' = a + (2tjr'' - 2i!r' + «r) 8c ;
and so on by successive substitutions.
(10) But any positive number |, rational or irrational, can be expressed
to any approximation desired in the scale of 2, as the radix of notation,
in the form
/3. + | + § + § + etc.;
where /9o is the integer next below f and )8i, )8j, etc. are either unity or zero.
If the series is finite any point of the form a ± ^Sc can be constructed
in a finite number of constructions ; if the series is infinite it can be con-
structed in an infinite number of constructions; and (since the series is
convergent) this means that in a finite (but sufficiently large) number of
constructions we can construct a point a ± fSc, where f *- f ' is less than any
assigned finite number however small.
Thus any point a + f c ou the line ac can be constructed, and similarly any
point on the line be can be constructed ; and it is sufficiently easy to see that
any point a + f i + i;c can be constructed.
This type of construction can easily be extended to a projective manifold
of any dimensions.
129. Grassmann's Constructions. (1) Grassmann's constructions*
in a complete region of two dimensions have for their ultimate object to
construct the point a + '^(f,, ^i)((ii'^<h)> where -^(fi, fs) is any rational
integral homogeneous function of fi, {2, and a, Oi, Og are any three given not
collinear points, provided also that the point a + {lOi + ^^^ and also certain
points of the form a + ttiOj + a^ are given, where the a's are known coefficients.
In order to accomplish this end the constructions are given for the following
series of points,
a-\-^i(ai + (h), a + f»(ai + flhi), a + fifa(oi + aa),
ct + f / (tti + a,) {where p is any positive integer}, a + fa" (aj + a,),
a + ^^ — ^' (Oi + Oa) {where 71 + 7« is not zero).
7i+7a
Then finally a construction is deduced for
when 7 + 7' -f- ... is not zero, and /a, v, jjf, v, ... are positive integers, and
/Lt + y = ft' + J/' = ....
* Cf. AusdehnungBlekre von 186*2, §§ 825—829.
220
DESCRIPTIVE GEOMETRY.
[chap. IV.
(2) Let a, a^, a^ denote any three elements forming a reference triangle
in the two dimensional region ; let a; = a + fiOi + f gOa', and let d = a + Oj + a,.
Fio. 4.
Firstly, make the following constructions (fig. 4),
yi = ajOa . ad, y^ = aroi . ad.
Then y^ = {aa^ + f lOiOa) (ooi f odg) = - (oaiaa) {a + f i (oi + Oa)}
= a + f 1 (tti + Oa).
Note that in future numerical factors which do not involve f i or fg will be
dropped without remark.
Similarly y* = a + f a (^i + «a).
(3) Secondly, make the following constructions (fig. 4),
Then e^ = (ooi + faOaai) («aa + Oa^i)
= — (oOiOa) tt — (aOiaa) Oq — ^^ (oaiO^) Oa
= a + Oi + fa^a.
And x' = (Gh« + facial) (oOa + f lOiOa)
= — (aOiOa) a — f 1 (oOiOa) Oj — f if g (aOiOa) Oj
= a + fia, + f if aOa.
And y = (ooi + f if aOatti) (a^i + oOa)
= (oOiOa) {a -h fif2(ai + Oa)} = a + fifa(ai + Oa)
= y^ • a^ . a . yiOa . O] . a(2.
(4) Now substitute yi for ya in the above construction (fig. 5). We
obtain
Vi = yi<h • Oad . a . yiOa . Oi . ad = a + f 1^ (aj + Oa).
129] qrassmann's constructions. 221
Similarly we construct
Fia. 5.
It is also obvious that in the constructions Oi and Og can be interchanged.
Thus
y = y^Os . Old . a . yidi . o^ . ocJ,
and yi' = yiO^ . OicZ . a . y^ai . o^ . ad,
and y2=y^'<hd' a .y^.a^-od-
Also in the construction in subsection (3) for y from yi and ^s, yi and y^
can be interchanged, thus giving two fresh forms of the construction, namely,
y = y^ai . a^ . a . y^ . Oi . ad, and j^ = yiO, . did . a . y^ . a, . ac2.
(5) Let the symbol (f i") stand for the point a + f i" (Oi + a,), and similarly
let (f/) stand for the point a + ^J^i^h +(^\ aiid let (f/f/) stand for the point
Now substitute the point (fi") for y, in the first construction given for
y, then we obtain
(f 1"+^) = (f 1") Ori . Oad . a . yiOa . Oi . (u2.
(6) Again, let pRi denote that the point p has been multiplied succes-
sively by the factors Oi, d^, a, yiO^, Oi, cid, so that j^jR^ stands for the point
pdi . Ojd . a . yiOs . Oi . ad In order to avoid misconception it may be men-
tioned that Ri is not the product a, . a^ . a . yiO^ .Ui.ad; for in pR^ the first
factor Oi is multiplied into the point p. Also pRi is itself a point : let
(pRi) Ri be denoted by pRi^, and so on.
222
DESCRIPTIVE QEOMETRT.
[chap. IV.
By applying this notation to the construction for (fi*"*"*) in tenns of f/,
we see that when v is a positive integer, (f i") = y^Ri^K
Since yia^ = xa^, R^ may be conceived to stand for the set of factors
a, , OgC^, a, xci^y Oi, ad successively multiplied on to a point. Also y^ = xa^ . ad.
Hence (f Z) = xa>^ . ad . i2i'^\
Thus the point a + f / {a^ + Oj) is exhibited as a product in which x occurs
V times.
(7) Similarly interchanging the suffixes 1 and 2, let B^ stand for the
set of factors Os, a^d, a, xa^^ a^, ad successively multiplied on to a point.
Also ya = ^^ • ^•
Hence (f /) = /cch . od . B^''\
Thus the point a + f,** (Oj -f- a,) is exhibited as a product in which x occurs
fi times.
Fia. 6.
(8) Again, in any of the constructions for y (say the first) substitute (f ,")
and (Is'') for y, and y,, say, for example, (f/) for y^ and (f/) for y^.
129] grassmann's constructions. 223
Then (f /f ^) = (?/) Oi . ctjd . a . (f i") a^^Oi.ad.
Hence the point a + ^i^j^ (o^ + Os) is represented as a product in which x
occurs (ji-{'v) times.
(9) Finally, let p and p' be any two points a-\-rff {a^-^-a^ and
a + «r' (oj + Oa) on the line ad ; and let c denote any point ^jOi — y^3^ on the
line aiO^. Make the constnictions (fig. 6)
q^pcti.pa^, r = qc .ad=pai,p'a2,c ,ad.
Then q = pdi . p'a^ = (aOiO,) {a + «r'ai + «ra,},
r = [yioch — 7jaaa + (^iw + 72iBr') a^] [acti + aa^}
= {aonq^ {(7i + 7a) a + (7i«^ + 72«^0 (^i + Oa)]
= a + ~ (Oi + Oa).
7i+7a
Similarly, let ^ denote a third point a + «r" (a, + a,), and let c' denote
the point (71 + 7a) Oi — 78^- Make the construction
r ^roi.pcti.c .ad=a+ — (oi + Oa).
7i + 7a+78
And so on for any number of points p, p\ p\ etc.
Thus if any number of points of the form (fi^f/) have been constructed,
then the point
can be exhibited in the form of a product.
(10) Hence the numerical product
v^*^) = ;;7z;7z (««i«9)-
7 + 7 + ...
It will be observed that x occurs in this product {(/i + i') + (/i' + j/') + ...}
times, and that therefore if j? be written in the form 97a + i/iOi + i^aOa, then the
product is a homogeneous function of 1;, i/i, i;, of degree {(/i -♦- v)-¥(ja +v) + ...}.
But let /} be the greatest of the numbers (/<& + i^),(/i^-f-i^^, etc., then it is
easily verified that the homogeneous function represented by the product is
any required homogeneous fiinction of degree p multiplied by 1; to the power :
[{(/A + 1') + (/i' + I'O "^ --•} " p]» ^^'^ ^I^ ^7 some constant numerical factor.
If however we keep x in the form a + f lOj + fjOa, then the most general
rational integral algebraic function (not necessarily homogeneous) of fi and
fa can be exhibited in the form of a product ; or if <^(fi, £^a) be the function,
it can be represented by a point
which is constructed as a product of the point x and fixed points, partly
arbitrarily chosen and partly chosen to suit the special function.
224 DESCRIPTIVE GEOMETRY. [CHAP. IV.
130. Projection. (1) Definition. Let the complete region be of v — 1
dimensions, and let x, y, etc., be elements on any given plane A of this
region. Let e be any given point not on this plane and let B be any other
given plane. Then the lines ex, ey, etc., intersect the plane B in elements
x\ y\ etc. : the assemblage of elements x\ y\ etc., on the plane B is called the
projection on B from the vertex e of the assemblage of elements /r, y, etc., on
the plane A,
Definition. Two assemblages of elements x, y, etc., on the plane A and
x\ y, etc. on the plane A\ which is not necessarily distinct from A, are
called mutually projective, if one assemblage can be derived from the other
by a series of projections.
If one figure can be deduced from another by a single projection, the two
figures are obviously in perspective.
(2) These constructions can be symbolized by products : thus the pro-
jection of a: on to the plane B from the vertex e is x'==xe . B. Let the
projected points always be assumed to be at intensities which are deduced
from the intensities of the original points according to this formula.
Since {\x + /ly) e.B = \xe . B + iiye . B, it follows that any range of ele-
ments on a line is projected into a homographic range.
(3) Proposition I. Let any subregional element in the plane A be
denoted by the product x^,x^..,Xf,, where p is less than i/ — 2; also let
a;/, x^ ... a?p' be the projections of the points x^, x^, ... Xp on to the plane B
from the vertex e, so that for instance a?/ = x^e .B ; then it will be proved
that*
x^x^ . . . a?p' = (eBy^^ XjX^ . . . Xffi . B = XiX^ . . . Xpfi . B.
In other words, if Xp be any subregional element of the pth order, and
Xf! be the corresponding subregional element formed by the projected points,
then
X; = (eBy-' X^ .B=Xffi,B.
Thus X^ will be called the projection of Xp, and the above equation forms
the universal formula for the projection of elements of any order.
(4) In order to prove this formula the following notation will be useful.
Let x^X2 ... (iP<r) ••• ^ifi denote that the elements rci, ajg ... ajp, e, with the excep-
tion of x„, are multiplied together in the order indicated. Then the extended
rule of the middle factor gives the transformations
XiX^...Xp/e.B^ 2 {'■'V]r'^{xaB)xTpo^..,{x^),.,Xffi'\-{'-Vf{eB)xiX^.,.Xf,,
Also d/p+i = Xp^^ e.B = (a?p+i B)e^ {eB) Xf^^ .
But X1X2... {xa) ...Xffi.e^Qy
ano X\X^ ... iXff) ... x^jb . Xp^^ ^ ^ x^x^ • • • {Xqi ... XtjXp^ie »
* This formula has not, I think, been stated before.
130] PROJECTION. 225
Hence by multiplication and rearrangement of factors it follows that
+ 2 {-V]r'^{eB){x^)xyX^...{x„) ...x^ie- {:-\y{eByx^x^...Xf^i
t«r»p+l -]
2 (- VjT'^ {XaB) x^x^ . . . (a?<r) . . . fl?p+i e + (- 1)^"*"^ {eB)xiX2 . . . iTp+i
= {bB) XiX^ . . . x^ie , B,
Hence by successively applying this theorem we deduce
Xi'x^' .,.Xp = ifiBY"^ X1X2 . . . Xfje . B = X1X2 . . . Xpfi . B,
(5) It is obvious that the relation between a point x and its projection
X is reciprocal ; that is, if a?' be the projection of a? on fi from vertex e, then
X is the projection of a?' on -4 from vertex e.
For afe.A = {(xB) e - (eB) x]e.A = '- (eB) xe . A
= (eB) (eA) x = x,
since (xA) = 0, by hypothesis.
Thus two figures are projective if they can both be projected into the
same figure.
(6) Proposition II. Any three coUinear points are projective with any
other three collinear points.
This is the same as the proposition that any two homographic ranges
are projective.
Firstly, let the two lines L and L\ on which the points respectively lie,
be intersecting, so that the complete region is of two dimensions. Let a, 6, c
and a', V, c be the two sets of three points on L and Z' respectively.
Take e and e' any two points on aa\ Construct the points eh.efV and
ec . e'c' : call them the points 6", c". Construct the point aa' . V'c" = a".
Then we have evidently
a = a^'e .X, 6 = V'e .X, c = c**e . i,
and of = ft'V . L\ V = V'e' . L\ d = c'V . U,
Thus the collinear points a'\ V\ c" can be projected both into a, 6, c and
a', 6', c'. Hence a, 6, c and a', 6', c' are projective.
(7) It may be noticed that if the regressive multiplications are defined
for a complete region of three dimensions and the ranges aho and aVc' be
coplanar, then the above results must be written
a = a"e . Ld, b = b"e . Ld, c = c"6 . Ld,
and a'=a"e'.L% V^V'e'.L'd, c' =- c"e\ L'd ;
where d is any point not in the plane of the straight lines L and L', and e and
e' being both on aa^ are in the plane of L and L\
w. 15
226 DESCRIPTIVE GEOMETRY. [CHAP. IV.
And more generally, if the regressive multiplication be defined for a
complete region of i^ — 1 dimensions, let D^, be any extensive magnitude of
the (i; — 3)rd order which does not intersect the two dimensional regions
containing L and L\ then LDt^^ and L'D^^^ can be taken as the planes of the
two projections, so that
a = a"e . LDy^^, etc., and a' = aV . L'D^^, etc.
(8) Secondly, let the lines containing a, 6, c and of a', V, d be not inter-
secting. Take any point 'p on ahc and 'p' on a'6V. Construct the line ^',
and on it take any three points a!\ h'\ c". Then a, 6, c and a", V\ d' are
projective, also a\ V, d and a", y\ c" are projective. Hence a, 6, c and
a', 6', c' are projective.
(9) Proposition III. If any p points in a subregion of p — 2 dimen-
sions (with only one addition relation) are projective with any other p
points in another subregion of p - 2 dimensions (with only one addition
relation), then any p-\-\ points (with only one addition relation) in a
subregion of p — 1 dimensions are projective with any other /» + 1 points
(with only one addition relation) in another subregion of p — 1 dimensions.
For let Oi, Oa; ... ctp+i and 6i, 6a, ••• 6p+i be any two sets of /» + 1 points in
regions of p — 1 dimensions.
Since the p + l points ai...a^i are contained in a subregion of /» — 1
dimensions, a^^i must intersect the subregion of the independent points
Oi, Oa, ... ap_i in some point c; and similarly hjl)^i must intersect the sub-
region of the independent points K K ... h^^ in some point d.
Now by hypothesis a series of projections can be made which transforms
6i, 6a ... 6p.i, d into diyO^.., ap^i, c. Assume that such a series transforms 6p
and 6p-|.i into b/ and 6'p+i.
Let Af^i stand for the subregional element OiO^... af^i, and let D^-p
denote the product of any v^p independent points which do not lie in -4.^-1,
where i/ — 1 is the number of dimensions of the complete region. Then
Af^i Dy^p is a planar element.
Again, c, 6p', 6'p+i are col linear and so are c, ap, ttp+i, hence 6p', 6'p+i, a^, a^^
lie in the same two dimensional region. Therefore ap6p' and ap+i6'p+i intersect
in some point e.
Let Dy^p be so chosen that e does not lie in the plane ilp_i2)^p: also let
D^p contain ap and ap+i. Then it cannot contain 6p^ and Vp^i^ since it does
not contain e.
Project on to the plane A^JJ^^p from the vertex e. The points
Oi, Oa, ... ap.1, c are unchanged, being already in that plane, also 6p' is
projected into dp, and 6'p+i into ap+i.
Hence the proposition is proved.
(10) It has already been proved that three collinear points can be pro-
jected into any other three collinear points ; it follows that any p points in a
180] PROJECTION. 227
subregion of /» — 2 dimensions are projective with any other p points in
another subregion of p — 2 dimensions.
(11) Proposition IV. The least number of separate projections required
can also be easily determined. For we have proved in the course of subsec-
tion (9) that if <f>(p) projections are required for two sets of p points in
subregions of /» — 2 dimensions, then ^ (p) + 1 projections are required for
two sets of p + 1 points in subregions of /> — 1 dimensions. We have there-
fore only to determine the least number requisite to project three collinear
elements a,b,c into three other collinear elements a, b\ d.
The construction given above in the second and general case may be
simplified thus. Join oh'. Project from any point e on hV. Then a is
unaltered, h becomes V and c becomes some point d^ Now project a, V d'
firom the point of intersection of oal and cV. Then a becomes a^ 6' is
unaltered, d' becomes d. Hence two projections are in general requisite.
Thus three projections are requisite for four points in a two dimensional
region, and p — 1 projections for p points in a region of p * 2 dimensions.
(12) These constructions still hold if the two sets of p points are both
in the same subregion of p — 2 dimensions. In such a case the same series of
projections which transforms one set of p elements into another set of p
elements may be conceived as applied to every point of the subregion. Thus
every point of the subregion is transformed into some other point of the
same subregion.
(13) Proposition V. It will now be proved that the most general type
of such a projective transformation is equivalent to the most general type of
linear transformation which transforms every point of the given subregion
into another point of that subregion.
If X be the point into which any point x is finally projected, the relation
between oi and x can be written in the form
X ^ X^\ • lS\ * 62 • -03 . ^3 . -^s * * * ^P^\ * -^p— 1 *
It is obvious therefore that x can be conceived as transformed into x' by
some linear transformation. The only question is, whether it is of the most
general type.
Now in the most general type of linear transformation, as applied to a
region of p — 2 dimensions, p — 1 elements must remain unchanged. Let
Oi, Os, ... ttp-i be these elements, and let any other point x be represented by
Then if a^ be the transformed element corresponding to Xy we have
a^ = ttifiOi + 09^/19 + ... + a^ifp_ia^i, where a^, Oj, ... a^i are the constants
which in conjunction with the fixed points define by their ratios the linear
transformation.
Hence if a given point ^ is to be transformed into a given point d, where
g^Xya, and c2=sSSa, we must have, aj = 81/71, 02 = 89/72, ... ap_i = 8p_i/7i>_i.
15— 2
228 DESCRIPTIVE GEOMETRY. [CHAP. IV.
Accordingly if the p — 1 unchanged points are arbitrarily assumed, it is not
possible by a linear transformation to transform more than one arbitrarily
assumed point into another arbitrarily assumed point.
But it is possible by a series of projections to transform the p points
Oi, Os, ... d^i, c into the p points diy ct^t ••• ^p~i> d.
Hence the general type of projection is equivalent to the general type of
linear transformation.
(14) It is to be noticed that a linear transformation can be conceived as
transforming all the points of the complete region of (say) i/— 1 dimensions.
But these points cannot be projectively transformed without considering
the region of i/ — 1 dimensions as a subregion in a containing region of p
dimensions. This fresh conception is of course alwajrs possible without in any
way altering the intrinsic properties of the original region of i/ — 1 dimen-
sions.
The Theory of Linear Transformation in connection with this Calculus is
resumed in Chapter VL of this Book.
CHAPTER V.
Descriptive Geometry of Conics and Cubics.
131. General Equation of a Conic. (1) The following investigation
concerning conics and cubics is in substance with some extensions a repro-
duction of Grassmann's applications of the Calculus of Extension to this
subject*. In places the algebra is handled differently and alternative proofe
are given for the sake of illustration.
A quadric surface in a complete region of two dimensions will be called
a conic. It will also in this chapter be called a curve in order to agree with
the usual nomenclature of Geometry.
(2) The complete region is of two dimensions: the product of three
points or of three linear elements or of a point and a linear element is
purely numerical. Also the product of three linear elements, being a pure
progressive product, is associative; thus if Xi, X,, X, be the linear' elements,
(XiXfXs) = (Xi . X,X,). Abo if p and q be points, then, since XjX, is a point,
(LiLipq) is the product of the three points XiX,, p, q. Hence
(LJj^pq) = (XiX, . pq).
(3) The equation, {xaBcDex) = 0,
where a, c, e are any points and B and D are any linear elements, is evidently,
since x occurs twice, of the second degree in the three co-ordinates of x.
For let ei, e^, e^ be the three reference points, and let x = fi^iH- fg02 + fi^s*
Also let the fixed points and lines be written in the form
and so on ; where oti, 0,, a,, etc., are given numerical coefficients. Then the
given equation, after multiplying the various expressions for the points and
lines, ti^es the form
Hence, dividing out the numerical fector ((h^^\ the given equation is
equivalent to a single numerical equation of the second degree defining a
quadric locus.
* Cf. AwdehnungtUhre von 1862, and CrelUt yoU. zzzi, zzzti, lii.
230 DESCRIPTIVE GEOMETRY OF CONICS AND CUBICS. [CHAP. V.
Write fT0 for the expression {xaBcDex\ then the following transformations
by the aid of subsection (2) are obviously seen to be true :
fSFx = [xaBcD . ex] = {xaBc (D . ea?)} = — {xaBc . exD]
= — {c . axB . exD] = {c . exD . axB] = — [xeDcBax] ;
where it is to be remembered in proving the transformations that xa \& ^
linear element, xaB is a point, xaBc is a linear element, xaBcD is a point.
(4) From cr^ = — (c . axB . exD] = 0, it is obvious that a and e are points
on the conic. In general c is not on the conic, for the points c, {cbcB), and
{ecD) are not in general coUinear.
(5) Points in which B and D meet the curve. Suppose that B meets
the curve in the point p, and let B=pq, Also substitute p for a? in the
expression Wg,
Now apB = ap ,pq = (apq)p = (aB)p.
Therefore «-, = - {c . apB , epD] = {aB) {cp . epD) = 0.
This involves either (i) that {aB) = 0, or (ii) that (cp . epD) = 0.
(i) Let {aB) = 0. Then axB = {aB) x - («?£) a = - {xB) a. Hence
'»■«= — (a?5) {ca . ftrD) = 0.
Therefore the curve splits up into the two straight lines
{xB) = 0, and {ca . exD) = 0.
Similarly if {eD) = 0, the curve becomes the two straight lines
{xD) = 0, and {ce . axB) = 0.
These special cases in which the conic degenerates into two straight lines
will not be further considered.
(ii) Let (cp . epD) = 0. Then {cp. cpjDj = jc(p.epi))}.
But p.epD=p {{eD)p - {pD) e} = - {pD)pe.
Hence {cp . epL) = ( pD) {cep) = 0,
so that p lies in 2) or in the line ce.
Accordingly the two points in which B intersects the curve are £ . D and
B.ce,
Similarly the points in which D intersects the curve are £ . D and D , ca.
(6) Letg==B.D,b^B,ce, d=^D.ca.
Then 6 = (J?6)c-(£c)e, d = (Da) c - (Dc) a.
Hence eb.ad=^ {Be) {Da) [ec.ac]:= {Be) {Da) {eac) c^c.
Also we may write B = bg, D = dg.
Hence the equation becomes
{{xa . bg){eb . ad) {dg . ex)] = 0 ;
where a, 6, d^e^g axe five given points on the curve and a? is a variable point.
(7) Conversely, if we take any five points a, b, d, e, g and write,
\{xa . bg) {eb . ad) {dg . ex)] = 0,
131, 132]
GENERAL EQUATION OF A CONIC.
231
then the above reasoning shows that the five points are on the curve which
is the locus of x. But only one conic can be drawn through five points;
therefore by properly choosing the five points this equation can be made
to represent any conic section, and is therefore the general equation of the
second degree.
(8) If we perform the constructions indicated by the products on the
left-hand side (cf. fig. 1), we see that the equation is a direct expression of
Pascal's theorem, which is thereby proved.
Fig. 1.
(9) Perform the operation of taking the supplement on the equation,
and write X for the linear element \x, A for \a, and so on. Then
I {(^a . bg) (eb .ad){dg. ex)} = {(XA . BO) {EB . AD) (DO . EX)]
= 0.
This is the general tangential equation of a conic [c£ § 107 (4)] : hence from
subsection (7) it follows that A, B, D, E, 0 are tangents ; and the equation is
a direct expression of Brianchon's Theorem.
132. Further Transformations. (1) These results can be obtained
by a different method which forms an instructive illustration of the algebra.
The following series of transformations follow immediately firom the
extended rule of the middle factor :
axB = {aB)x — (xB)a;
hence, axBcDx = {{aB) xc.D- (xB) ac.D]x
= (aB) (xD) ex - (xB) (aD) ex + (xB) (cD) ax.
Now (aB)(xD)-(xB)(aD)=^x[(aB)D-(aD)B] = x[a,DB]^(xa.DB).
Hence, axBcDx = (xa . DB) ex + (xB) (cD) ax,
and, (axBcDxe) = (xa . DB) (cxe) + (xB) (cD) (axe).
232 DESCRIPTIVE GEOMETRY OF CONICS AND CUBIOS. [OHAP. V.
Thus the equation of the curve, Wx = 0* can be written
(xa . DB) (cxe) + (xB) (axe) (cD) = 0.
(2) To find where B meets the curve, put (xB) = 0. Then either
(xa . DB) = 0, or (cxe) = 0.
Thus either x is the point BD or it is the point ceB ; therefore these are the
points where B meets the curve.
Similarly the points where D meets the curve are BD and caD.
(3) Obviously the points a and e lie on the curve.
(4) If (cD) = 0, the curve degenerates into the two straight lines
(xa . BD) = 0, (xce) = 0.
Similarly if (cB) = 0, the curve becomes the two straight lines,
(xe . BD) = 0, (xca) = 0.
(5) To find the second point in which any line through the point a cuts
the curve.
Let L be the line, then (aL) = 0. Let x be the required point in L, then
xa = L,
Hence (xaBcDex) = (LBcDex) = 0.
Hence x is incident in the linear element LBcDe, also x is incident in L,
Therefore x = LBcDeL.
(6) It is to be noticed that a apart from L does not appear explicitly
in this expression for x. Hence the theorem can be stated thus :
If a be any variable point on the line L, the conic through the five
points a, BD, ceB, e, caD passes through the fixed point LBcDeL.
(7) The conditions that T should be the tangent at a are (aT) = 0, and
a = TBcDeT.
(8) The general expression Wg. is susceptible of a very large number
of transformations of which the following is a type :
xa.bg =: (xhg) a — (abg) a?, c6 . ad = (ehd) a - (eba) d, dg.ex = (dgx) e — (dge) x.
Hence {(xa . bg)(eb . ad)(dg . ex)] =(eba)(dge)(xbg)(adx) - (eba)(ade)(xbg)(dgx)
-h (abg) (ebd) (dgx) (aex) — (abg) (eba) (dgx) (dex),
(9) The equation,
(xoiB^aJS^ . . . an-iBn^ia^x) = 0,
represents a conic. Hence the following theorem due to Grassmann :
*If all the sides of an n-sided polygon pass through n fixed points
respectively, and n — 1 of the coiners move on n — 1 fixed lines respectively,
the nth comer moves on a conic section.'
133, 134] LINEAR CONSTRUCTION OP CUBICS. 233
133. Linear Construction of Cubics. The first linear constructions
satisfied by any point of a cubic were given by Grassmann* in 1846 ; and
the theory was extended and enlarged by him in 1848 and 1856 f. An
indefinite number of such linear constructions of increasing complexity can
be successively written down by the aid of the calculus. The simplest tjrpes
are given by
(xaAoi . xbBkCbi . xc) = 0 (1),
(axtAori .xbBbi .xc)= 0 (2),
(xaBcDxDicB^a,x)^0} ^ ^'
(xaA.xbB.xcC) = 0 (4).
The two equations, marked (3), are alternative forms of the same equa-
tion. It is to be noted that none of these constructions give a method of
discovering points on a cubic; but that, given a point ^ on a cubic, the
constructions can be made. Thus a point x on the cubic will be said to
satisfy the corresponding construction, but not to be found by it.
134. First Type of Linear Construction of the Cubic. (1) To
investigate the construction
(xaAoi . xbBkCbi . xc) = 0.
This equation asserts that if the three lines xaAoi, xbBkCbi, xc are
concurrent, the locus of ^ is a cubic. Let y be the point of concurrence ;
then the construction is exemplified in figure 2.
Fzo. 2.
(2) It has now to be proved that any cubic can be represented by
this construction. This will be proved by shewing that by a proper choice
* Cf. CrelWt Journal, vol. xxxi.
t Cf. CrelU^t Journal, vols, xzxvx. and ui.
234 DESCRIPTIVE GEOMETRY OF CONICS AND CUBICS. [CHAP. V.
of the fixed lines and points of the construction the cubic may be made
to pass through any nine arbitrarily assumed points. Thus we proceed
to investigate the solution of the following problem : Given any nine arbi-
trarily assumed points in a plane to find a linear construction satisfied by
any point of the cubic passing through them.
But previously to the direct solution of this problem in § 135 some
properties of the expression (xaAoi . xhBkCbi . xc) must be investigated.
(3) Let mx stand for the product {xdAa^ . xbBkCbi . xc).
Then cxaj = — (xdAa^ . xc . xbBkCb^).
Now put p = xaAoi .xc, y = xbB.
Then «»•* = - (l? . qfcCbi) = (pbiCkq).
It is easily proved that (pbiCkq) = — (qkCbip),
(4) To find the particular positions of x for which p = 0, or g = 0.
Now p = 0, when x = a, and when x=c.
Also by § 106 p = (xaAoic) x — (d»a-4.aia?) c
= {(xA) (ooic) — {aA) (x(iic)] x - (xA) (odix) c.
Hence all the points x for which p = 0 (except x^c) must satisfy (in
order to make the coefficient of c zero) either (xA) = 0, or (xdOa) = 0.
If (xA) = 0, then, since the coefficient of x must also be zero, (xaic) = 0.
Hence x = aicA ; and thus OicA is another of the required values of x for
which p vanishes.
If (a?aai) = 0, then (a;il)(aaic) — (a^)(a^c) = 0. The only point on the
line aoi which satisfies this equation is the point a. For if Xa-^fuii be
substituted for x, the equation reduces to fi (a^A)(aa^c) = 0 ; and hence, /a = 0.
Hence the three values of x for which p^O are a, c, OficA.
The only value of x for which g = 0 is a? = 6.
(5) To investigate the values of x for which p = x. These are included
among the points satisfying the equation px = 0. Though this equation for x
is also satisfied by the points just found which make jd = 0.
Now px = -- (xA) (aoix) ex.
Hence if x lie in A, Le. if (a?il) = 0, or if x lie in aai, i.e. if (a?aai) = 0,
then p = x. But the points a and aicA must be excluded, as involving p = 0.
(6) The points for which q = x are given by qx = 0, excluding the
point b for which g = 0.
Now qx = xbBx = (xB) xb.
Hence if (xB) = 0, then q^x.
Thus either of the points AB or aa^B substituted for x in the expressions
^ and? make p^x^q.
136] FIRST TYPE OF LINEAR CONSTRUCTION OF THE CUBIC. 235
(7) Hence if or be either of these points
^x = (phiCkq) = (xbjCkw),
Now (xbiCkx) = (xbik) (xC),
Therefore (xbiCkx) = 0, implies either (xbik) = 0, or (xC) = 0.
Hence if the points AB and aOiB lie on the cubic they must lie either on
bjc or C Thus it A, B, Che concurrent, the point of concurrence lies on the
cubia
This analysis of the equation will enable us easily to follow Qrassmann's
solution of the problem.
136. Linear Construction of Cubic through nine arbitrabt
POINTS. (1) Let the nine given points be a, b, c, d, 6, /, g, h, % ; and let
the cubic be (xaAoi . xbBkCbi . xc) = 0. Then the curve obviously goes
through the points a, b, c.
Let dicA, which lies on the cubic [c£ § 134 (4)], be the point d; and let
A, B, C he concurrent in the point e, which is therefore on the cubic by
§ 134 (7). Hence we may write A=de,
Let the point aoiB lie on bik and therefore be on the cubic by § 134 (7) :
let it be the point/, so that (Jihk) = 0. Hence both e and/ lie on B ; there-
fore we may write B^ef, Also OicA = d now becomes (iiC,de = d; hence d
is the point of intersection of aiC and de and therefore (OiCcJ) == 0. And
(OfiaB) =/ becomes Oia . ef=f; hence (oio/*) = 0. Therefore, since Oi lies both
in cd and in af, we may assume Oi = af, cd.
It has been assumed that no three of a, c, d, e,/, are collinear ; for other-
wise some of these equations become nugatory.
(2) We have k, (7, 6] still partially at disposal: the conditions to be
satisfied by them being only,
(Ce) = 0, {fbjc)^0.
Now let gi = gaAoi .go^ga (de) {of .cd).gc,
hi s= haAoi ,ho^ha (de) (af, cd) . he,
%i^iaAai.ic^ia(de)(a/.cd).tc,
gt^gbB =gb.ef,
h^^hbB ^hb.ef,
i^ =5 ibB ^ih.ef.
Thus the six points ^i, Ai, ii, g^, A,, i, can be obtained by linear construc-
tions from the nine given points a, 6, c, ... i. We proceed to choose k,C,bi,
so that the following equations hold [cf § 134 (3)]
(gJcCb,g,)^0, (hJcOnhd^O, (iJcCb,%0^0,
286 DESCRIPTIVE GEOMETRY OF CONICS AND CUBICS. [CHAP. V.
(3) Let 0 and k be chosen, if possible, to satisfy the equation
(ixftiOH,) = 0
without conditioning hi. Then for this purpose we must write iJbiC = ii ; that
is to say, C must be assumed to pass through iy. But e lies in (7, hence we
must assume C=^eii.
Hence k is given by (khii) = 0. Further, except in the special case in
which (/eii) = 0, k and bi are also [cf. subsection (1)] related by (fkbi) = 0.
Thus k = iii^,fbi, where bi is as yet any arbitrarily assumed point.
(4) The remaining equations can be written
(kg/Jgybi) = 0, {kh^ChJ>i) = 0.
Hence k must also be such that the three lines kf, kg/Jg^, kh^Chy intersect in
the same point bi ; also k lies in ii^. Therefore k is one of the points in
which lit, intersects the cubic curve,
{xf. xgJOgi . xhiChi) = 0.
(5) But this curve is formed of three straight lines. For if x be any
point in (7, then
xg^C = X, xhjO = x^
and hence {xf. xgfig\ . xhJOh^ = {xf, ayi . xhi) = 0.
Thus C is part of the locus.
Now g2{=gbB) and h^ (= hbB) both lie on B, also / lies on B. Thus if x
be any point on B,
xf^B, xgjOgi = BCg^, xh^Oh^ = BCh^,
Hence {xf . xgJOg^ . xhjOhi) = {B . BCg^ . BCh^) = 0.
Thus B is part of the locus.
(6) Hence the remainder of the locus is another straight line.
To find this required line, let y = xf. xgjOgi,
Then (y . xhfih^ = {xhfih^) = — {yhiCh^x).
Hence the equation of the three straight lines is
{xf. xg^Ggi . xh^Chy) = [xf{xg/Jgy) hfih^x] = 0.
This equation is satisfied by any value of x for which
xf{xgfig^ Ai = 0 ;
that is, by any value of x making xf and xgfig^ intersect in h^ ; that is,
if X satisfies {xfh^ = 0, and {xgfigji^ = 0. But {xg^GgJi^ = {hygiCg^x) ; hence
X must lie in the intersection of fhy and hqgiCg^. Therefore
X ^ higiCgt . fhi.
Similarly another point on the third line is g\hiCh^.fgi. Hence the
required line which completes the locus is
{hgfig^ 'f>h) {gACK fgi)-
185] LINEAR CX)NSTRUCnON OF CUBIC THROUGH NINE ARBITRARY POINTS. 237
(7) Put K^{Kgfig^.fh,){gAGKJgi\
Then k must lie in iit, [by subsection (3)] and m B or C or K,
Now the assumed equation of the cubic is
{xaAoi . xbBkCbi . xc) = 0.
Assume that k lies in B. Then id>Bk = {aibk) B.
Hence the equation of the cubic becomes
(xbk) {xaAoi . BCbi • ^ = 0.
Accordingly the cubic degenerates into the straight line hk and a conic
section ; and cannot therefore be made to pass through any nine arbitrarily
assumed points.
Assume that k lies in (7. Then xhBkC = (xbBC) k.
Hence the equation of the cubic becomes
(xbBC) (xaAoi . kb^ . xc).
Thus in this case also the cubic degenerates in a conic section and a
straight line, namely, BCb.
Therefore the only possibility lefb is that k lie in K. It will be shown
that this assumption allows the cubic to be of the general type by showing
that the cubic passes through the nine arbitrarily assumed points.
Hence let it be assumed that k 33 iiiJB^.
Accordingly with these assumptions the equations
(ffACkg,) = 0, (AAGA:A,) = 0, (iJ>,Ck%,) = 0,
are satisfied.
Again, 61 has been determined, for by subsection (4) it is the point of
intersection of
kf, kg^Cgu khfih^\
hence bi = kgjOg^ . kf.
(8) Finally, therefore, it has been proved that the equation,
xaAdi . xbBkCbi . rrc = 0,
denotes a curve passing through the nine arbitrarily assumed points a, b, c, d,
^yfy g, K h provided that A, B, C, Oi, b^, k are determined by the linear
constructions,
A^de, B^^ef, O^s^ei^, a^^af.cA, k^iiij^, bi^kg^Cgi.kf;
where
gi==^ga(de)(af, cd).gc, hi^hob{de){af, cd),hc, ii = ia (de) (af. cd). ic,
g^-gb.ef, h^^hb.ef, h^ib.ef,
and if = {KgiCg^ -A) (5^ACA« .fg^.
(9) This linear construction satisfied by any point x on the cubic repre-
sents the general property of any ten-cornered figure a?, a, 6, c, d, e, f, g, h, %,
inscribed in a cubic. It is the analogue for cubics of Pascal's Theorem for
conies.
238
DESCfilPTIVE GEOMETRY OF CONICS AND CUBICS. [CHAP. V.
136. Second Type of Linear Construction of the Cubic. (1)
Equation (2) § 133, namely
(xaAoi . wbBbi .xc)=0
is a simplified form of (1), which has just been discussed. It can be derived
from (1) by putting k = bi. For in this case
ta>Bb,Cb, = {a^BbJ)^) C - (C60 xbBbj
= ''{Ob^)xbBbi = a:bBb^.
(2) Hence as in § 134 (4) and (7) the points a, b, c, OicA, AB, aoiB are
seen to lie on the curve.
Similarly, from the symmetry of the equation, bicB, bbiA are seen to be
points on the curve.
Also it is easy to see that aoi . &&i is a point on the curve.
Let these nine points be denoted by a, 6, c, d, e,f, g, h, k respectively ; so
that
d^OicA, e^ABf f^aoiB, g^bjcB, h^bbiA^ k = aai,bbi,
(3) To prove that the cubic denoted by this equation is of the general
type.
Take any cubic, and inscribe in it any quadrilateral khef as in the
figure 3.
Fig. 8.
Let the side kh meet the curve again in b, the side he meet the curve
again in d, the side ef in g, the side fk in a. Assume c to be any other
point of the curve not collinear with any two of the other points. Then
136, 137] SECOND TYPE OF LINEAK CONSTRUCTION OF THE CUBIC. 239
the assumed points on it determine the cubic. Join cd cutting ^in Oi, and
eg cutting hk in bi. Then iffe = B,hs^A, the equation
xaAui . a^Bbi . ^c = 0,
has been proved to represent a cubic through the nine points. Hence by a
proper choice of constants the equation can represent anj cubic.
(4) The construction represented by this equation is exemplified in
figure 4.
Fig. 4.
137. Third Type of Linear Construction of the Cubic. (1) The
equation (3) of § 133 is
tj. = (xaBcDxDiCiBiOix) = 0.
The points a and Oi obviously lie on the curve.
To discover other points on the curve, notice that by § 132 (1)
axBcDx = {xa . DB) ex + {xB) (cD) ax.
Hence tj, = {xa . DB) {cxD^CiByxa^ + (ocB) (cj)) {axD^CyB^xa^ (A).
But (cxDiCiBiXOi) = 0 is, by § 131 (4) and (5), a conic through the five
points c, Oi, BiDi, CiOi^u (HQBi.
Also (axDiCiBixa^) ^ 0 is, by § 131, a conic through the five points a, a,,
J5iA, Ci^iAi CiaBi.
Hence the points AA ^nd CiOiDi lie on both conies and therefore also on
the cubic.
But tj, = xa^BiCiDixDcBax.
Hence BD and caD are also points on the curve,
240 DESCRIPTIVE GEOMETRY OF CONICS AND CUBIC8. [CHAP. V.
(2) As a verification notice that, i{ x^BD, then
xaB = Xy xaBcD = xcD = Xy xaBcDx = xx=:0:
also, if a? = ca . D, then
xa=ca, xaBc = caBc=ca, xaBcDx=caDx = xx^O.
(3) To find where D cuts the curve a third time; note that axBcD
is a point in D ; hence if x be also in D, axBcDx = D, excluding the case
when axBcDx is zero.
Hence, by substituting D for axBcDx in the equation of the curve, we
see that x satisfies (DDiCiBiOix) = 0, and (Dx) = 0 : therefore x = DDjCiBiaiD.
Hence D cuts the curve in the three points BD, caD, DDiCiBiOiD, and
similarly A cuts the curve in the three points 5iA> CiOiD,, DiDcBaDi.
(4) The two conies {cxD^c^Byxa^ = 0, and {axDiCiBxa^ = 0, have been
proved to intersect in the three points Oi, BiA> CiOiA- The fourth point
of intersection is caAciA^i - oa ; since by § 132 (5) this is the point in
which the line ca meets either conic.
Hence the three points in which the line ca meets the cubic are a,
caDy caDiCiBiOi . ca. Similarly the three points in which the line c,ai meets
the cubic are o^, CiOiA) c^a^DcBa . dOi.
(5) It is easy to obtain expressions for the three points in which the
line BDa cuts the cubic. Two of the points are already known, namely, a
and BD. To find the third notice that from equation (A) of subsection (1),
the required point is the point, other than a, in which the line cuts the conic
(xaD^c^B^a^x) = 0. By § 132 (5) this is the point {BDa) D^c^B^a^ (BDa), which
can also be written
BDaD^c^ByO, . BDa.
Thus the three points in which BDa meets the cubic are a, BD,
BDaDxCiB^Oi . BDa. Similarly the three points in which BJ)ia^ meets the
cubic are dy, BJ)^, BiD^a^DcBa . B^DiO^.
(6) To find the three points in which B cuts the curve, notice that
if (xB) = 0, then fi*om equation (A) of subsection (1) the equation of the
curve reduces to
(xa . DB) (cxDjCiBixai) = 0.
Hence either xa . DB = 0, and x = BD, which has been already discovered;
or {xcDiCiBidyx) = 0. Therefore the two remaining points in which B meets
the cubic are the points in which B meets the conic {xcDyCiB^dyx) ^ 0.
These points can be immediately expressed, it B = Bi, In this case the
cubic becomes
(xaBcDxDiCiBaix) = 0 ;
and it will be proved [c£ subsection 13] that the equation still represents any
cubic curve.
137] THIRD TYPE OF LINEAR CONSTRUCTION OF THE CUBIC. 241
The points where B meets the conic, (xcDjCiBoix) = 0, have been proved
in § 131 (5) to be BDi and cCiB. Hence B meets the simplified cubic in
the three points BD, BJ)i, cc^B.
(7) The transformation
tr^ = — a? . xaBcD . xOiBiCiDi
is established as follows.
Let Xi = xaBcDxDiCiBi, then tj, = ajiOiic = — iCi . a^Oj ; since the product
of three points is associative.
Let X2 = xaBcDxDiCi, then tj^ = — Z2B1 . aroi = Xa . aJOifii ; since the pro-
duct of three linear elements is associative.
Let 07, = xaBcDxDi, then iit„ = x^Ci . xOiBi = — ic, . xoiB^d.
Let Xi, = xaBcDx, then «r. = — X4D1 . xOiBiCi = X^ . xOiBiCiDi .
Hence «r. = (xaBcD) x . xoiBiCiDi = — a? . xaBcD . xa^B^cJ)^,
The previous results can be easily obtained by means of this form of
the equation.
(8) The geometrical meaning of the equation is that x, xaBcD, and
xOiBiCiDi are collinear. This property is shown in the annexed figure 5.
jC^Di
Hence if two variable triangles have a common variable vertex, and
two sides, one of each triangle, which meet in the common vertex lie in
the same straight line, and if also the four remaining sides pass respectively
through four fixed points, and the four remaining vertices lie respectively on
four fixed straight lines, then the locus of the common vertex is a cubic.
(9) The four lines A (= ca), D, A^ (= dOi), A have a special relation to
the cubic
(xaBcDxDiCiBiOix) = 0,
in addition to the fact that the points caD and CiOiA both lie on the curve
[cf. subsection (1)].
For suppose that the lines A, D, Au A are arbitrarily assumed. Then
the points AD (= e) and A^D^ (= d) are determined.
. w. 16
242 DESCRIPTIVE GEOMETRY OF CONICS AND CUBICS. [CHAP. V.
Also suppose that the remaining points in which A and A^ cut the curve
are arbitrarily assumed on these lines, namely [cf. subsection (4)],
/(= ADiC^BiOiA), /i(= AiDcBaAi), a and Oj.
Thus a, Oiffyfi are supposed to be known, and the equations /= ADiCiBiOiA
and /i=^AiDcBaAi partially determine Ci and Bi, and c and B, which are
the remaining unknowns.
Again the arbitrarily assumed lines D and Di are supposed to meet the
curve in two arbitraiily assumed points e(= AD) and e, (=-4.iZ)i)- Let two
other points k and ^i in which D and J)i respectively meet the curve be
arbitrarily assumed, so that [cf. subsection (3)] we may assume
k = DD^cBa^J), and h, = D^DcBaD^,
Then the remaining points in which D and A respectively meet the
curve are [cf. subsection (3)] BD and B^D^. Call these points g and g^. It
will now be shown that g and g^ are both determined by the previous
assumptions of the eight points a, Oi, 6, ei,/,/i, A, A^i; and that accordingly
the group of four lines A,D, A^, D^ must bear some special relation to the
cubic curve which passes through the eight assumed points.
(10) For if Li and L^ are linear elements and pi and pa are points, the
extended rule of the middle factor gives,
LiL^Pi = (iiPi) L^ - (Api) A, and pipJL^ = ( piXJp, - {pjj^)p^.
Remembering these formulae we see that
/oi = AD^CiB^OiAoy = — {Aa^ AD^cJB^Oy = AD^dB^Oi ;
fa,B, = AD,CiB^aiBj^ =^ - {a^B^) AD^o^B^ =AD^c,B^;
foiBiCi = A DiCiB^Ci = — (fiiCi) A D^Ci = ADiC^ .
Hence ADid • /^i . -Bj = 0.
Similarly DAci .ka^.B^^O.
Hence B^ passes through the points ADiCi ./oi (=p) and DD^Cj . Arch (= g).
Therefore we may write Bi = (-4 A^i ./oi) {DD^Ci • ^^i) =/>?.
Hence ^r, = AA = (^ Aci . /o^) (i)Aci . Araj) A = p?A = (pA) ? - (?A)i>.
Now (pA) = (il Aci .fa, . A) = - (il Aci . A . M)
= (C A) (^ A . A) == (ci A) {AD^a,).
And (jA) = (DAci . fcoi . A) = - (DAoi . A . ha,) = (Ci A) {DDJca,).
Hence gri = (AD^fa^) q - (DDJca,) p.
But p = ilAci -A = (^-Oyoi) cx - {cJu,)AD,;
and g = DD^c, . ka, = {DDJca^ Ci - (ciArOi) i)A.
Also (cjih) = - (ill/), and (cikoi) = - (^lA:) by subsection (9).
Thus g, = (A,k) (AD Jo,) DD, - (A J) (DDM) AD,.
137]
THIRD TYPE OF LINEAR CONSTRUCTION OF THE CUBIC.
243
Hence the position of gi is completely determined by the arbitrarily
assumed elements.
Similarly the position of ^r is completely determined.
(11) Hence ten points on the cubic are now known. The cubic is there-
fore independent of the positions of c and Ci on A and A^ ; except that c
must not coincide with a or AD, nor Ci with a^ or -4iAi in which cases some
of the previous equations become nugatory.
(12) We will now prove that the specialized form of equation introduced
in subsection (6), namely
(waBcDxDiCiBaix) = 0,
where (cA) = 0 = (ciD) represents the most general form of cubic.
The three points in which D cuts the curve are [c£ subsection (3)], BD,
caDf DDiCiBdiD.
But since (c^i)) = 0, BD^CyBa^D = DBaJ) = DB,
Hence D touches the curve at BD and cuts it again in caD. Similarly
A touches the curve at BD^ and cuts it again in CjOiA*
Also [cf. subsection (6)] B cuts the curve in the points BD, BD^, cciB,
Fia. 6.
(13) Now (cf. fig. 6) take any cubic curve and draw the lines D and A
tangents to it at any two points g and gi. Join ggi by the line B which cuts
the curve in another point h. Through h draw any line cutting D in Ci and
A in 0. The tangents D and A cut the curve again in two points e and ^.
16—2
244 DESCRI^IVE GEOMETRY OF CONlCS AKD CUBICS. [cHAP. V.
Now join ec; this line cuts the curve in two points. Call one of the two a.
Similarly call one of the two points, in which SiCi cuts the curve, Oi.
Then by construction h = cCi-B, e = caD, Ci = CiaiDi-
Now the tangents D, A at g and gi and the points A, e, ^i, a, Oi completely
determine the cubic.
But (xaBcDxDiCiBoix) = 0 is a cubic satisfying these conditiona Hence
this equation represents the assumed cubic.
138. Fourth Type of Linear Construction of the Cubic. (1) The
equation (4) of § 133 is
(xaA . xbB . xcC) = 0;
and it represents the fact that the points xaA, xbB, xcG are collinear. The
construction is shown in figure 7.
It will be shown that any cubic can be thus described.
Fig. 7.
(2) To find where A cuts the cubic, note that if x lies in A, xaA = x.
Hence (xaA . xbB . xcC) = (x . xbB . xcC) = (xB) xb . xcC
= {xB) (xC) (xbc) ;
where the sign of congruence means that only constant factors have been
dropped.
Therefore the three points in which A cuts the cubic are AB, AC, bcA,
Hence by symmetry, BC, caB, ahC also lie on the cubic. Also obviously
a, 6, c lie on the cubic. Thus the two triangles respectively formed by a, 6, c
as vertices, and hy A, B, C as sides are both inscribed in the cubic and their
corresponding sides, namely A and be, B and ca, C and ab, intersect also on
the cubic.
,i.
ooi;'
il
138] FOURTH TYPE OF LINEAR CONSTRUCTION OF THE CUBIC. 245
(3) We have to prove that, given any triangle abc inscribed in a cubic,
a triangle A, B, G always exists with these properties relatively to a6c and
the cubic
Take a, b, c any three points on a given cubic, not collinear. Let he cut
the cubic again in/, ca in g, ah in h.
Let a, 6, c be the reference triangle, and let f, ^, 5" be the co-ordinates
of any point x. Then we can write a? = f a + ^6 + fc.
Let Ay B,G be any straight lines through /, jr, A. Then, since any
numerical multiples of A, B, and C can be substituted for them, we may
write il = \6c + y^ca + fiiob,
B = yjbc + fica + a^b,
C = fij>c + ctjca + vab ;
where \, fi, v are at our disposal and )8i, 71, 7j, Oo, )8s, as are known from the
equations, /= be Ay g = caBy h = abC and from the fact that one of the letters
with each subscript can be assumed arbitrarily without affecting anything
except the intensities of Ay By 0, which are immaterial.
Now auA = (xA) a — (aA)x
= (abc) {(Xf + y,v + AD a - X (fa + ^6 + fc))
= (abc) f(7ii; + A?) « - X^6 - Xfc}.
Similarly xbB = (abc) {— fi^a + (Oaf + y^) b — /tfc},
xcC = (a6c) {- i/fa — vrjb + (Af + Ojiy)}.
Hence, (a?a-4. . a?6B . xcC) = 0 can be written as the ordinary algebraic
equation,
-i/f, -prjy Af + Osi;,
This becomes on expanding the determinant
(W + A?) («2?+ 72?) (Af + «,i7) - /ti;^? (7,17 + A?)
- >'>^?f (oa? + 72?) - V^7 (A? + 0,^) - 2\,ip^v^= 0.
This is the equation to a cubic through the six points a, b, c,/, g,h: it is
required to determine X, fi, v so that it may be the given cubic through
these points.
The given cubic is determined by any other three points on it fi, gi,hi
forming another triangle. Now X, /a, v can be so determined that the above
equation is satisfied by the co-ordinates of these points. For by substituting
successively the co-ordinates we find three linear equations to determine
X, fi, Vy each of the form
jt 1 -tTa JLm JLa ■»-»
X fi V Xfiv
where Pj, P,, ... , Pj do not contain X, fi, v.
246 DESCRIPTIVE GEOMETRY OF CONICS AND CUBICS. [CHAP. V.
Now put <r for \fw, and solve these three linear equations for X"^ /a~^ i/"*.
Then we may assume
1 Hi j^ 1 H.2 Y^ 1 H^ r.
X a fi (T pa
where Hi, J?a, .- J^s do not contain \ fi, v.
Hence multiplying and replacing Xfiv by a, an equation of the form,
is found; where Po, Pi, P2> Pz do not contain X, fi, v. Thus there are three
values of a, one of which must be real. Hence there are three systems of
values of X, fi, v ; and one system must consist of real values. Thus three
systems of values can be found for A^ B, C; and one of these systems must
make A, B, C tohe real lines.
Thus three triangles, of which one must be real, can be found related to
a, b, c and to the given cubic in the required manner.
Let A, B, C he one of these triangles. Then
(xaA . xbB . xcC)=i 0
is the given cubic.
The above proof of the required proposition is different from that which
is given by Grassmann *.
139. Chasles' Construction. (1) Another construction for a cubic
given by Chaslesf, without knowledge of Grassmann's results or methods is
represented by
xeDpEdF . xfB . xdC = 0,
where {Ff) = 0 = (Bdy
(2) It is easy to prove J the following relations:
The points d, 6,/, BG, OF lie on the curve.
The third point in which de cuts the curve is
deDpEdF{deC)Bf{de).
The third point in which ef cuts the curve is
e/DpEdF (e/B) Cd (ef).
The third point in which BCfcnts the curve is
FCdEpDe (BGf).
* CrelUf vol. lii.
t Comptes Rendust vol. xxzvi., 1863.
t Gf. Grassmann, loc. cit.
139] CHASLES' CONSTRUCTION. 247
The third point in which BCd cuts the curve is
BFdEpDe(BCd).
Also if we put a = deDpEdF(deC) Bf(de\
h = efDpEdF(efB) Gd(ef), c = FO, A = BCf,
a, = cdEpDeAc(de), bi^ BFdEpDeBc(ef)]
then the given cubic can be expressed by the construction
xaAoi ' xhBbi . arc = 0.
CHAPTER VI.
Matrices.
140. Introductory. The leading properties of Matrices, that is of
Linear Transformations, can be easily expressed by the aid of the Calculus
of Extension. A complete investigation into the theory of Matrices will
not be undertaken in this chapter: the subject will only be taken far
enough to explain the method here employed and prove the results required
in the subsequent investigations in the theory of Extensive Manifolds.
Orassmann treated the subject in his Avsdehnungslehre of 1861 apparently
in ignorance of Cay ley's classical memoir on Matrices*. An exposition and
amplification of Grassmann's methods was given by Buchheimf. The
present chapter is in its greater part little more than a free translation of
Orassmann's own writing, amplified by the aid of Buchheim's paper ; except
that Qrassmann and Buchheim do not explicitly consider the case of a matrix
operating on an extensive magnitude of an order higher than the first ; and
that the treatment here given of symmetrical matrices is new, and also that
of skew matrices. I have also ventured in § 146 to distinguish between
latent regions and semi-latent regions: in the ordinary nomenclature both
would be called latent regions.
141. Definition of a Matrix. (1) Let 6i, e, ... c,, be any v reference
elements in a complete region of i/ — 1 dimensions. Let the symbol
Cfr] , U<2 ... (^p
v\ , &2 • • * ^p
prefixed to any product of some or all of these elements, be defined to
denote the operation of replacing the element ei by Oj, the element e, by a,,
and so on. Thus if 6^, ex ... Cp be any of the original reference elements,
CL\ , Uj • * • ^K
— - _ ^K^k ••• ^P ^ 0/K^X .•• Gtp>
* Phil. Trans, vol. cxlviii., 1868 ; and ColUeted Mathematical Papers, vol. n., no. 162,
t Proc. London Math, Soc. vol. zvi. 1886,
140, 141] DEFINITION OF A MATRIX. 249
80 that in this instance the symbol of operation has transformed the product
e^\ . . . gp into the product a^y, ,,,af,. Let ^ be put for the symbol — ^ — '-^ — ^ .
The convention with respect to the operator ^ will be the same as that with
respect to the operator | which is stated in § 99 (9). Then </>«i = Oi, ^63 = Oj,
<^i^ = diO^i and so on.
(2) It follows from this definition that ^ is distributive in reference to
multiplication. For ^i^e, = a^a^ = ^^1^, and so on.
(3) Furthermore let <}> be defined to be distributive in reference to
addition, so that if Ei, E^.,,Ef^ be regional elements of the crth order
formed by the multiplicative combinations of the crth order formed out of
the reference elements [cf. § 94 (1)], then
<j^ (pLiEi + OqE^ + • • • + ^pEf^ = oii^Ei + ag^i^s + . . . + oip><l>Ep,
For example, if a? = fi«i + fje, + ... + f,^„, then
= ftcti + f aOj + . . . + f „a,,.
(4) The operator ^ — called by Qrassmann a quotient — may be identified
with Cayley s matrix. For assume
Then ^ = (a„f , + a^f 2 + . . . + a,„f ^) e,
+ (Onfi + Oaf* + ... + oi^^y)e^
+
+ (a^fi + ^i^fj + . . . + (Kv^v) e^'
Hence if we put <f>x = rj^ei -{- rf^ , . , +Vt^^f tt^^J^ with the usual notation
for matrices,
(1;,, i;, ... 7fp) = ( ttu, a,2 ... a,^ 1^ f,, f, ... f^).
Og, OCss .«• ^if
My] , ^^ ... ^irp
Thus we may identify ^ with the matrix Ha^
(5) It will be convenient to call the elements Oi, 0^ ... a^ the elements
of the numerator of the matrix, and 6], 63... 6,, those of the denominator.
The elements of the denominator must necessarily be independent, if the
matrix is to have a meaning.
250 MATRICES. [chap. VI.
142. Sums and products of Matrices. (1) If E^ denote any regional
element of the ath order, say eie^ . . . e^, then
E„ = X^OiOa . . . tta = X'^ - — = E^
If <f> denote the matrix —^ — — — ^ , then the matrix — - — " will
be said to be the matrix <;^ multiplied by \, and will be written symbolically
X<l>, This convention agrees with the ordinary notation, and will cause no
confusion when the matrix is operating on elements of the first order, but
must be abandoned when the matrix operates on regional elements of order
greater than unity.
(2) If two matrices, operating on v independent elements of the first
order, give the same result in each case, then they give the same result
whatever extensive magnitude they operate on.
For let Oi, Ca...c„ be any v independent elements, ^ and % the two
matrices.
Assume 4>Ci — x<hy ^9 = X^«' ••• ^^"^X^*"
Then any extensive magnitude i4^ of the <rth order can be written as the
sum of terms of which XCiCa ... c^ is a type. Hence
Thus the two matrices ^ and x i^^^^ ^^ considered as equivalent, and their
equivalence may be expressed by <;^ = %.
(3) If Ci, Cg... C|, be any v independent elements, and if the matrix <f>,
originally given as — '- '-^ — - , give the results 0Ci = di, ^Cg = dg . . . <^r = d^^
V\ , ^2 • • • &p
then <f> can also be written in the form — - — '-'- — - . For if A be any exten-
C\ , Cg ... Cy
sive magnitude, it follows that <I>A =—^ '-^^—^ A.
Hence any matrix can be written in a form in which any v independent
elements form its denominator.
(4) The sum of numerical multiples of matrices operating on any element
of the first order can be replaced by a single matrix operating on the same
element. For it can be seen that
{
CUi , CZg ... Cvp ^ Oi y 0^ ... 0^
a hp h...
C\ , ^2 ■ * ■ ^y ^1 i ^2 * * * ^v
> X = X,
€i , ©2 » • • •
But if the extensive magnitude operated on be of order greater than the
first, then this theorem is not true. For example consider the product eie^
Then
1
Ol , dg . . . O] , O) . . . r , L L
^ Z ^ ^ — I ' ^^ = ^^9 + Ma-
&^ , &] « • ■ ^If ^2 * * *
142] SUMS AND PRODUCTS OF MATRICES. 251
But in general aia^+ih&2 is not a single force, and cannot therefore be
derived &om eie^ by the operation of a single matrix.
(5) A numerical multiplier can be conceived as a matrix. For if
a? = 2fc, then Xa? = 2f\6, where X is some number. Hence \ may be con-
ceived as the matrix — *-^ — " .
If A^ be an extensive magnitude of the <rth order, then
X^i, X^2 • • • X6,» . w A
C\ f 6% • . • Op
Also from subsection (4) if <;^ be any matrix, \ any number, and w an
element of the first order, then (<f> + \)x can be written ;^a?, where % is a
single matrix.
(G) Let <f> and x ^^ ^^^ matrices and A any extensive magnitude.
Then the expression <l>x^ is defined to mean that the transformation x^ = ^
is first effected and then the transformation <f>B,
The combined operation <l>x can itself be represented by a single matrix.
For let €1,62 ,., e^he the independent reference elements, and let
and let ^Oi = 61 , (fxi^ = 62 . . . ^a, = b^.
Then the matrix which replaces ei by tj, eg by 6a ••• ^^ by b^, is equivalent to
the complex operation ^.
(7) The operator <^, when operating on an element of the first order,
may be conceived as a product [cf § 19] of two matrices. For let -^ be a
third matrix, and a any extensive magnitude of the first order. Then
Hence the two operators <f>(x-^'*k) *^^ ^X + ^^ *^^ equivalent.
It is to be noticed that the sum of the matrices is another matrix and
the product of matrices is another matrix. It will be convenient to speak of
the product of two matrices when the matrices are operating on a magnitude
of an order greater than the first. In this case the matrices have not a sum
[cf. subsection (4)], and therefore strictly speaking have not a product
[cf. § 19].
The product of three matrices is associative ; that is ^ . y^A = ^ . x^-^«
For the meaning of (f>x • '^A is that a single matrix <})i is substituted for the
product <fyx, and the meaning of <;^ . xi^-^ ^ ^^^^ ^ single matrix Xi is substi-
tuted for the product ^'^; and then the equation asserts that <l>i^A =■ <f>XiA.
Now let ^1, ^s . . . a„ be the v reference elements ; then, taking a t3rpical
element only, let i^gp = a^, x^9 = ^p> ^p = ^p- Hence
^ittp = Cft, and j^i^p = 6p.
252
MATRICES.
[chap. VL
Therefore <t>i^^p = ^^p ~ ^p» ^^^ ^Xi^p "^ ^p =" ^p*
Thus ^-i^Cp = ^Xi^p ; and, since this is true for every reference element,
<f>iy^A = <l>XiA, where A is any extensive magnitude.
143. Associated Determinant. If the matrix ^ can be written in
the alternative forms — ^ — —'— " and - - — — '— ^, then the ratios -. ~~~\
and ^ ^ are equal.
For let Ci = 711^1 + 712^1 + . . . + 7ik«,», with i; — 1 other similar equations.
Then d^ = ^, = yn<l>ei + 712^2 + . . . + 711.^^1. = yu(h + 7ij^ 4- . . . + 711/1.. with
1/ — 1 similar equations.
Hence (ciCa ... c„) = A (^i^a ... e^), where A stands for the determinant
7iii 7" ••• Yi" I •
7«i» 7m ••• '/«»'
71^1 » 7r2
>!'
Similarly (d^d^ . . . d„) = A (ajOa . . . a„).
Finally therefore, (d,d,...d,) ^ (a^,^^
(2) If with the notation of § 141 (4) the matrix be
V ^11 > ^li • ■ • ^iv ) I
I 0^> ^ ••• C^
then the ratio (aiO^ . .
I ^vl > ®r2 • • • ^V¥
c^k)/(^i^ . . . c,,) is the determinant
^llj ^U ••• ^iv
Offl, Otj2 ... OEs,,
^n > 0^1^ . . . Ot
¥¥
144. Null Spaces op Matrices. (1) If the 1/ elements which form
the numerator of a matrix are not independent, so that one or more relations
exist between them, then the matrix can always be reduced to the form in
which one or more of the elements of the numerator are null.
For let the matrix <f> be — * — — -; and let Gi, a, ... a^ be independent,
while the remaining v-^fi elements of the numerator are expressible in
terms of the preceding ft elements ; so that we may assume v fi equations
of the form
a^+p = ttpiOi + oCpfl^ + . . . + Upftfif^,
143, 144] NULL SPACES OF MATRICES. 263
Let Cft+i , Cfi+2 . . . Cr be defined by (v — /a) equations of the type,
where i>dti+p = a|*+p.
Then it is easily seen that ^i, 63 ... 6^, c^+i, c^+a ••• c„ are v independent
elements. Hence these elements can be chosen to form the denominator of
the matrix.
Hence the matrix takes the form — — — - — — .
(2) In this case the associated determinant is zero. The region
of v — fi'-l dimensions, is called the null space of the matrix ; and the
matrix is said to be of nullity p — fi. Thus if the associated determinant
vanish, the matrix is of nullity other than zero.
Any point in the null space is said to be destroyed by the matrix, and
will be called a null point of the matrix. Any point x is transformed by the
matrix into a point in the region («i, e^... e^^. This region (ei, «» ... e^ is
said to be the space or region preserved by the matrix.
(3) Sylvester* has enunciated the theorem that the nullity of the
product of two matrices is not less than the greater of their nullities, but
not greater than the sum of the two nullities. The following proof is due to
Buchheimf.
Let ^ be a matrix of nullity a and let Na be its null space ; and let % be
a matrix of nullity /8 and let N^ be its null space. Also let P^^ and P„_^
be the spaces preserved by ^ and x respectively. Then if JV. and P^^
intersect in a region 2a of 8 — 1 dimensions, the nullity of the matrix ^ is
/9 + S. For to find the nullity of ^ we have only to find the most general
region which x transforms into Ta, since any point in this latter region is
destroyed by ^. Now if Tj and N^ be taken as co-ordinate regions [cf.
§ 65 (3)], any point in the region of /8 + S — 1 dimensions, defined by the
co-ordinate elements lying in Ti and N^, is transformed by ^ ^^^ * point
in 2a. Thus the nullity of ^x ^^ ^ + ^» ^"^^ ^^ ^^^^ space of ^ is the
region defined by the co-ordinate points lying in T^ and N^. Hence the
nullity of ^ is not less than /9, being equal to /8 if JV« and P^_^ do not
intersect. Also it is immediately evident that the nullity of ^ is at least
equal to the nullity of ^: for if x^ ^i® ^ -^«» *ben ^a? = 0. Hence the
nullity of ^ is not less than a.
* Gf. S Johm Hopkiru Circulan 83, " On the three Laws of Motion in the World of Universal
Algebra."
t Cf. Phil. Mag. Series 5, yoL 18, November, 1884.
254
MATBICES.
[chap. VI.
Again, to prove that the nullity of (f>x is less than a + fiy note that if
a-{- 0>v the theorem is obvious. For a matrix of nullity v would destroy
all space. Assume therefore a + fi<v. Now B is greatest when Jf. is con-
tained in P„-.^, since a<i/ — /8; hence the greatest possible value of 8 is a.
Thus the greatest possible value of the nullity of (f>x is a + /3.
(4) Buchheim extends Sylvester's theorem. For if a + (i/ — /8) < i/, that
is, if a < /8, then in general Na and P^-p do not intersect. In this case there
is no region Tg. Thus if a <i8, the nullity of ^;;^ is in general /8. Again, if
a + {v — /3)>p, that is, if a>/3, then in general Na and Py-p intersect in a
region of a — /8 - 1 dimensions ; thus b = a- fi, and in general the nullity of
<f>X is a.
Thus in general the nullity of (f>x is equal to the greater of the two
nullities of ^ and j(^ ; but if special conditions are fulfilled, it may have any
greater value up to the sum of the two nullities.
146. Latent Points. (1) If a point x is such that, <f> being a given
matrix, a^ = p/p^
then X is called a latent point of the matrix, and the ordinary algebraic
quantity p is called a latent root.
The transformation due to the matrix does not alter the position of a
latent point x, it merely changes its intensity.
(2) Let the latent point x be expressed in the form X^e. Also let
{(f>-'p)ei = Ci, (^ — p)«, = c„ and so on.
Hence (<^ — p) a? = 0 = 2f (<^ — p) c = 2f c.
Therefore Ci, c^... c, are not independent, and thus (CiCt ... c,) = 0.
This equation can also be written 11 [(<^ — p) g] = 0, that is
{(<^ei - pei) (^, - pe^) . . . (</>c^ - pe^)] = 0.
This is an equation of the i/th degree in p, of which the first term is
(— iy(ei ... 6„)p'', and the last term is {(f>ei . <^, ... ^„). The roots of this
equation in p are the latent roots of the matrix.
(3) From § 142 (4) and (5) with the notation of § 141 (4), ^ - p is the
matrix
( au — p, ttia ... ai„ ).
^ J «a2-T-p ... flay
l,rl
, a.
... U|fif ^^ fj
Hence the equation for the latent roots is
ttu — Pi ctij ... ai„
a
yl
Ors
Opy — p
= 0.
146] LATENT POINTS. 255
(4) If all the roots of this equation are unequal, then p, and only v,
latent points exist, one corresponding to each root, and these points form an
independent system. These propositions are proved in the following three
subsectiona
(5) There is at least one latent point corresponding to any root />, of
the equation giving the latent roots. For let
Then since (CiCj . . . c„) = 0, a relation holds such as yiCi + 7aCj + . . . + 7yC„ = 0.
Hence 7i(</>-/3i)«i + 7j(</>-/)i)ea + ... +7,,(</>-pi)e„ = 0;
this becomes ^ {716, + 7,^4 + . . . + 7„6„} = pi {7161 + 726, + . . . + y^^]-
Hence the point 7161 + 78^2+ ... -{-y^y is a latent point corresponding to
the root pi of the equation.
(6) A system of v such points, one corresponding to each root, form a
system of independent elements. For let Oi, ct, ... a„ be the v latent points;
then, if they are not all independent, at least two of them are independent,
otherwise the v points could not be distinct.
Assume that the fi points Oi, a, ... a^^ are independent, and that another
point a, can be expressed in terms of them, by the relation
a^ = fltiOi + Ogas + . . . + GCiAd/A.
Then <^^ = ai^Oi + ffa^NXa •+• . . + a^4^|A»
that is, patta = CLipiOi + 0^2^ + . . . + OitiPiiflii'
Multiply the first equation by p^ and subtract from this equation, then
0 = (fh - Pir)aiai + (p2 - Po) ««a, + ... + (p^- Pa)*^^!*-
Since none of the latent roots, pi, pi ... p^ are equal, this forms one relation
between Oi, a^... af^ contrary to the h}rpothe6i& But at least two of the
latent points must be independent, hence they are all independent.
(7) Two latent points cannot belong to the same latent root. For
assume that Oi and di' are two distinct points such that ^Xh^pi^hi (fxii' = piOi'.
Let 0^,09... a, be latent points corresponding to the remaining 1/ — 1 roots.
Then oi, a, ... a, form an independent system. Hence a,' can be written in
the form aiOi + a^^ + ... + CLpd^.
Hence <f>ai' = ai^Oi + oii<fxii + ... + a^ipa,
— Pi^i<h. + PiOf^ + . . . + p/t/tp*
But <fxii' =s piOi = piGCidi + PiOCsOs + . . . + piOLpdv
Therefore (p2 - pi) «A + (p» - /h) «»a, + ••• + (pr - Pi) a^ar = 0.
Accordingly there is a relation between a2>as...a,., which has been
proved to be impossible.
Hence there is only one latent point corresponding to each latent root,
when the latent roots are all unequal.
256 MATRICES. [chap. VI.
146. Semi-Latent Regions. (1) Let the region defined by fi latent
points of a matrix with unequal latent roots be called a semi-latent region
of the (ji - l)th species. The number indicating the species of a semi-lat^it
region is thus equal to the dimensions of the region when all the latent
roots are unequal.
(2) Let Ci, eg ... e^ be the v latent points of a matrix with unequal latent
roots pi,p2'"pv' Then the region defined by ^i, e,... e^(/Lt< i/) is a semi-
latent region. The characteristic property of a semi-latent region is that if
X be any element in the region, then ^ is an element of the same region ;
for if a? = fi^i -I- . . . + Siifi^9 then
And if X be any regional element incident in the semi-latent region,
then (f>X is a regional element incident in the same semi-latent region. In
particular if Z = A^ea • • • «m> ^^^^
(f>Ij = \piP% . . . piifiiB^ . . . 6|A = PiPa • • • Pfi-"'
(3) It is also important to notice that
<l>X = piX + (pi - pi) fa^a + . . . + (p^ - Pi) itfii^
= pxX -h x\
where a?' is a point in the semi-latent region eg ... 6|a> excluding ei. Thus x'
lies in a semi-latent region of the {p, — 2)th species, whereas x lies in a semi-
latent region of the (/* — l)th species.
147. The Identical Equation. (1) If pi, pa ... p„ be the latent roots
of a matrix, no two being equal and none vanishing, and if Oi, a,... a,, be
the corresponding latent points, then it follows from above that the matrix
can be written in the form
PlOl, p^ ... PjjOy
Qri , Ctj ... Qfp
(2) If <^ be the matrix, let <^* denote the matrix <^<^, <^* the matrix <^<^<^,
and so on. Also any point x can be written fiOi -h ^^ + . . . + f /i^.
Hence 4>x = pif i^i + PafiAj + . . • + p^iwf^w •
Hence <f>x- piX = (j)2- pi) faOa + ... + (pir — pi) fra„.
Again <^ (<^ — pi^?) = (f>^X - pi<^ = pa (pa - pi) fa^a + • • • + p., (p,r — pi) fr^ir.
Hence (fPx - (pi -|- pg) </>a? + pip^ = (p, - pa) (pj - Pi) fjOj + • • •
+ (Pk - Pa) (p.' - Pi) frflir.
Proceeding in this way, we finally prove that
(<A - Pi) (<A - P2) ••. (<A " Pi')^ = 0,
whatever element x may be.
146 — 148] THE IDENTICAL EQUATION. 257
(3) The equation may be written
(<A - Pi) (0 - /^) — (<A - P") = 0,
that is, </>" - (pi + Pa + ... p^) 0*^^ + ...(- iypiP2 ... Pk = 0.
This is called the identical equation satisfied by the matrix <}>. A similar
equation is satisfied by any matrix, though the above proof has only been
given for the case when all the roots are unequal and none vanish.
148. The Latent Region of a repeated latent root. (1) In the
case when the equation giving the latent roots has equal roots, assume
that tti of the roots are pi, Og are ps, ...a^ are p^, where pi,p2'^»Pii are
the fi distinct roots of the equation. Then
(2) Then subsections (5) and (6) of § 146 still hold, proving that at
least one latent point corresponds to each distinct root, and that the /i latent
points which therefore certainly exist are independent.
(3) Consider now the root pi which occurs ai times, where a^ is greater
than unity.
Let eiy 62... Bphe V reference elements, and for brevity write
PiCi — <\>ei = €1, pifig — <^e2 = eg', ... piOp — 0e„ = c/.
Then since pi is a latent root of the matrix ^, {e^e^ ... ej)=^0. Hence
Ci', e^ ... ej are not independent [cf § 145 (2)].
(4) Assume that i' — A of them and no more are independent, so that
there are /81 relations of the type
A«a^ "h A«o2^ -h . . . + K0^^ = Vl \^)»
where a is an integer less than or equal to /81 and equation (1) denotes the
crth relation of that type.
Let Oia = X^nCi + Xo^^s + • • • + X^K^r-
Then {pi - <^) Oj^ = Xnfii + ... + \,^J = 0.
Hence <^a = P\<h9 •
Thus corresponding to each relation of the type (1) existing between
ei', e^ ... ej, there is a latent point, such as Oia, corresponding to the root p^.
Hence, since /81 relations have been assumed to exist, there are ^1 latent
points of the type 0^9. Furthermore all these points are independent. For
if not, the relations of the type (1) are not independent.
(5) The region of A — 1 dimensions defined by Ou, Ou ••• ^h^, is such
that if X be any point in it, <\> (x) = piX.
This region is therefore such that every point in it is a latent point,
corresponding to the root pi. Let it be called the latent region of the
matrix corresponding to pi.
w. 17
258 MATRICES. [chap. VL
(6) The number /8i cannot be greater than ai. For let Ou, Ois... o,^,,
defining the latent region corresponding to p,, be chosen to be /8i of the
reference elements 61, e,... 6^. Thus let a,i = ei, an^e^ and so on. Let e^
stand for (p'-<f>)e,, e^ for (pi'-<f>)ea. Then the equation (eigj... c^) = 0,
contains the factor (p — pi)*'.
But (p — <l>) ttia = (p — )t)i) ttia = (/o — Pi) «tr, when <r ^ ft.
Hence the equation becomes (p — pi/» {6162 . . . 6^,^^l+l ...«„) = 0.
Therefore ySj < aj.
149. The first species of semi-latent regions. (1) If /8i<ai, then
(^162 . . . ^^,^^^+l . . . e„) = 0, is satisfied by the root p^ which occurs ai — /81 time&
Hence (61^2 • • • ^fififi+i • • • ^/) — 0.
Thus the v elements ^i, 62... e^,, e'^+i ... c/ are not independent. It is
known that the ^1 elements ei, e^ ... e^^ at least are independent. Assume
that 1^ — 71 only are independent (1/ — 71 > fii). Then 7^ relations hold of the
type
where a is put successively equal to 1, 2 ... 7,.
Since e],^...6^, are independent, in each relation of type (2), all the
coefficients /itr, Pi+i -^ P'o.v cannot vanish, nor can it be possible to eliminate
all the elements e'^^^x ... ej between these relations and thus to find a
relation between ^i ... e^^.
Thus if we assume 71 elements of the type
then these elements of the type 61^ are mutually independent, and are also
independent of 6, , eg . . . «^, . Also
The coefficients ^« ... /c^p^ cannot all vanish: for otherwise 6,,^ would belong
to the latent region corresponding to pi, which by supposition is only of
/81 — 1 dimensions.
Let a\a stand for K^nCi + ... «l^^^c^,, then a\a is a point in the latent region
corresponding to pi : and
<f>bi9 = pA» + a'lir.
(2) Thus 7i independent elements, b^, ... biy^, satisfying an equation of
this type [cf. § 146 (3)] have been proved to exist, defining a region of 7, — 1
dimensions. Also by the same reasoning as in § 148 (6) it is proved that
(3) The fii independent points of the latent region of the type cLia
corresponding to the root pi and the 71 points of the type 61^, just found,
together define a region of ^1 + 71 — 1 dimensions, which will be called the
149, 150] THE FIRST SPECIES OF SEMI-LATENT REGIONS. 259
semi-latent region of the first species corresponding to the root pi. This
definition is in harmony with the definition of semi-latent regions given in
§ 146 for the case where all the latent roots are unequal. For let sc be
any point in this semi-latent region, then 00? is another point in the same
region ; let X be any regional element incident in this region, then <f>X is
a regional element incident in the same region. And if Z be a regional
element denoting the semi-latent region itself, then <\>L = p^^'^y^ L. Also we
can write, ^ = piX-\-y, where y belongs to the latent region (that is, to the
semi-latent region of the zero species).
It should be noticed that by definition the semi-latent region of the first
species corresponding to any given repeated root contains the latent region
corresponding to that root.
(4) The region defined by the points of typical form 6ja in subsection (1)
is contained within the region defined by e^,+i ... 6„ ; while the latent region
is defined by ^i, Ca ... e^^. Hence the region defined by the points
6,a(<r = l, 2...71)
does not intersect the latent region.
But from subsection (1), ^^ = pA, + a'l^, where a',^ lies in the latent
region. Now it can be proved that the 71 points a\aW~^> 2 ...71) are
independent. For if a relation of the type, Sfo^'ia = 0, holds between them,
then by writing (0 — pi) hi, for a\oy we have
2f a (<A - pi) hi, = 0, that is {<j> - pO l^ahi, = 0.
Hence the point 2f Aa lies in the latent region, and therefore the region
defined by Sia(cr = l, 2... 7,) must intersect the latent region, contrary to
what has been proved above.
Hence the 71 points a'ia(o- = 1> 2 ... 71) are independent. But they all
lie in the latent region which is defined by ^1 points. Hence ^1 ^ 71.
150. The higher species of semi-latent regions. (1) Semi-latent
regions of the second and of higher species can be successively deduced by
an application of the same reasoning as that of § 149 (1).
Thus to deduce, when /8i-h7i<ai, the semi-latent region of the second
species, corresponding to the repeated root pi, take as before ^i of the refer-
ence elements, namely 6i, CJ...e^^, in the latent region, which is assumed
to be of A - 1 dimensions, and take 7, of the reference elements, namely
in the semi-latent region of the first species (but not in the latent region),
so that the i8i-h7i reference elements thus assumed define the complete
semi-latent region of the first species. Then, if <r ^ A, ^ir = Pi^ir, and hence
ip-<l>)e,=^(p-pi)e,. Also, if cr>A and ^ A + 71, ^a = />i«^ + air, where
a„ lies in the latent region and is therefore dependent on Ci, e^.-.e^^. Hence
(p-<f>)e, = (p-pi)ea''a,.
17—2
260 MATRICES. [chap. VI.
Thus {{p - <l>)e, . (p - <l>) e^ ... (p - <l>) e^,+yJ = (p - piY'-^' e^e^... e^^+y^.
Hence the equation for the latent roots, namely, 11 {(p — <^)6a} =0, can
a«i
be written {p - p^'^'*' [BiC^ . . . e^,+Yl ^ {(P " ^) ^»}] = ^'
<r=P,+y,+l
But the equation for the latent roots has by hypothesis the factor (p — p,)*',
where A + 7i < «n thus the expression
V
[«i«2...6^.+y, n [{p - (f>) e^]]
<r=^,+y,+l
contains the factor (/o — /Oi)"' ~^' ~^'.
Thus writing pi for p we see that the v points
are not independent.
Assume that v — Si only ai'e independent (i' — Si > /8i + 71), so that there
are Si independent relations of the type
- {A*<r,^,+y,+l (Pi - <t>) «^,+y,+l + • • • + A^crr (pl - </>) «4 = ^ >
where a is put successively 1, 2 ... Sj. All the /a's cannot vanish simul-
taneously in any typical relation ; and all the terms of the type
(Pi-<A)«t(t>A + 7i)
cannot be simultaneously eliminated between the Si relations, so as to leave
a relation between the independent elements Cj, ^a ... ^^,+Y,•
Now assume c^, = /L^a.^,+Y,+^«^,+Y^+l + • • • + Z**^."
Also note that the point tctn^-^ k^A •¥...+ Ka^p^+yfip^+y^{=bi^ ssiy) lies in
the semi-latent region of the first species. Hence the above typical relation
takes the form
<f>0l9 = PlOlif + ^a.
(2) Also by the same reasoning as in § 148 (6), it follows that
A + 7i + Si = «i-
(3) Also by the same reasoning as in § 149 (4) it follows that the region
defined by the Si points Ci»(<r = 1, 2 ... 81) does not intersect the semi-latent
region of the first species. Also, as before, the 81 points of the type bio are
independent and the subregion defined by them (lying in the semi-latent
region of the first species) does not intersect the latent region ; for otherwise
some point of the type SfiaCur lies in the semi-latent region of the first
order, contrary to the assumption that this semi-latent region is only of the
/81 + 7i — 1 dimensions. Thus 81 ^ 71.
(4) If fii^tti, then only a latent region exists corresponding to the
repeated root a^ and no semi-latent region. If ^, < Ui and /81 + 71 = ai, then
no semi-latent region of a species higher than the first exists. If /9i + 71 < ai,
and /9i + 7i + 81 = tti, then no semi-latent region of a species higher than the
151, 152] THE HIGHER SPECIES OF SEMI-LATENT REGIONS. 261
second exists. If A + 7i + 8i < oti, then by similar reasoning a semi-latent
region of the third species exists, and so on till ai independent points have
been introduced defining the complete series of semi-latent regions corre-
sponding to the root pi.
Also from subsection (3) and from § 149 (4) it follows that if a^ > fifii, where
fi is an integer, then in addition to the latent region at least > semi-latent
regions of the successive species must exist*.
(5) It follows from (3) and § 149 (4) that a matrix can always be written
thus
^a> ^19 f (ha > •••
where only those typical terms are exhibited which correspond to the latent
root pi.
161. The Identical Equation. (1) Suppose that the number of
different groups of points of the types Oi^, 6ia, Cia, and so on corresponding
to a latent root pi is Ti. Then
and if pia be a point in the Xith group, that is in the semi-latent region of
the (ti — l)th species (but not in that of the (t — 2)th species), then
(<^ — pi)^'Pi^=0. Let the region defined by all these points be called the
semi-latent region of the matrix corresponding to pi.
(2) Now all the points of the diflferent types thus found, corresponding
to all the latent roots, are independent, and may be taken as a reference
system.
Hence if Tj, t, ... t^ be the corresponding numbers relating to the other
latent roots, and x be any point, then
{<t>-Piy'{<l>-p,r...(<l>-pj^x=o.
Thus any matrix satisfies the identical equation
(<^-p,r(0-^r-...(0-p^)v=o.
(3) Since t, < oti, Tj < etg . . . t^ < a,*, it follows that any matrix satisfies the
equation
(.^ - p,r (^ - p,r • ■'(<!>- pm)"" = 0.
Thus the equation of § 147 is proved for the case of equal roots. But in
this case the matrix satisfies an equation of an order lower than the i/th.
162. The Vacuity of a Matrix. (1) A null space [cf. § 144] can
only exist if a matrix has a zero latent root. The null space, or null region,
iff the latent region corresponding to the zero latent root.
* This theorem does not seem to have been noticed before : nor do I think that the relations
Yi < /3j, d] ^ y^, etc. have been previously explicitly stated.
262 MATRICES. [chap. VI.
(2) If the zero latent root occur a times, then the matrix is said to be
of vacuity a. Thus by definition the vacuity of a matrix is not less than its
nullity. Let the semi- latent regions corresponding to the zero root be called
also the vacuous regions of the matrix. Thus if 6 be a point in a vacuous
region of the first species, <^ = a, where a is a point in the null region ;
also if c be a point in the vacuous region of the second species, <l>c=^b,
where 6 is a point in the vacuous region of the first species ; and so on.
(3) Assume that S independent points, and no more, can be found in
the vacuous regions of the first species defining a subregion which does not
intersect the null region. Let cfi, c^^ ... (2a> be these points, and let the /3
points bijb^.^.b^ define the null region. Then any point x in the vacuous
region of the first species can be written 2f d + Siyi.
Also by §150(3), S</3\ and we may assume consistently with the
previous assumptions, ^ = Xifej, (fxi^ = \J>2, ... (f>d$ = Xj)^.
Hence ^a; = <^2f d + <^S7;6 = 2f ^ = 2 ^f^pbf,.
P=i
Thus any point in the vacuous region of the first species is transformed into
a point in the subregion of the null region defined by 6i, fcj ... 6a- Call this
subregion the subregion of the null region associated with the vacuous
region of the first species.
163. Sybimetrical Matrices*. (1) In general, if a; and y be any two
elements and (f> any matrix, {x\<l>y) is not equal to (y |^).
In order to obtain the conditions which must hold for these expressions
to be equal, let the matrix be — '- — *-^^ — - , where, according to the notation
of § 141 (4), ttp = ttipCi + otaipea + ... + a„pe„.
In other words the matrix is ( au, a^, ... ai„ ).
(
Then, supposing that e^, €2, ... e^ are a set of normal elements at unit
normal intensities [cf. §§ 109 (3) and 110 (1)],
(6p I (t>ea) = (^p I a,) = a^ (e^ j^p) = a^,
and (ea \ </>0 = (««r | Op) = a^p (e^ \ e^) = cVp.
Hence, if the required condition holds, ttp^ = a^fft.
(2) Thus the matrix with the desired property is a matrix symmetrical
about its leading diagonal when the elements of the denominator form a
* Symmetrieal matrices are considered by Grassmann [cf. Atudehnungslehre von 1862, § 891] ;
but his use of supplements implicitly implies a purely imaginary, self -normal qaadrio. Hence
his couolasions are limited to those of subsection (7).
153] SYMMETRICAL MATRICES. 263
normal system (at unit normal intensities) with respect to the quadric chosen
as the self-normal quadric.
Let such matrices be called symmetrical with respect to the normal
systems, or, more shortly, symmetrical matrices.
(3) If /A out of the V latent roots of a symmetrical matrix be distinct and
not zero, so that at least /i points Ci, Cs, ... o^, can be found with the property
4>c^ = 7|/;p, then the /i points Ci, Ca, . . . c^ corresponding to different latent roots
7i> 7s> ••• 7fi are mutually normal.
For let a7 = fiCi + f2Ca + -.. + ffiC,t, and y = ^iCi+i7sCa+ ... +Vi»Pn^-
Then (y | <Aa?) = (2i;c | Sfyc) = 2 (fpi7a7p + fai7p7a) (c^ M,
and • (x I (f>y) = (2f c | Xvyc) = 2 (f p^,7» + f ^^pTp) (^p I c*).
Hence (y i </«:) = (a; | <l>y) gives 2 (fpi7a - ^cVp) (7p - 7a) (Cp | c^) = 0.
Now let all the f's, except fp, and all the 17's, except 17^, vanish ; and it
follows that (Cp \cc) = 0.
Hence Ci, Cg ... c^ are mutually normal.
(4) Let Ci\ Ci", etc., be other points in the latent region of the root 71, so
that ^' = 7iCi', etc.: then the same proof shows that c/ is normal to all of
C3, ... c^, and so on. Hence the latent region corresponding to 71 is normal
to the latent region corresponding to p,, and so on.
(5) In the same way it can be proved that the whole semi-latent region
corresponding to any latent root 71 is normal to the whole semi-latent region
corresponding to any other latent root 72. For let di be any point in the
semi-latent region of 7, of the first species.
Then <f)di — 7,(^1 -h XjCi, ^c^ = 7,02.
Hence (ca | ^) = 71 fe jrf,), by (3) and (4).
Also {di \^) = 72(di |C2) = 7a(Ca |di).
But (Ca I ^di) = (di I <^), by hypothesis. Hence (71 — 7a)(ca|di) = 0, and
7i+7a by hjrpothesis. Therefore (cg |di)=0. Hence the semi-latent region of
the first species corresponding to 71 is normal to the latent region correspond-
ing to 72. Similarly the semi-latent region of the first species corresponding
to 78 is normal to the latent region corresponding to 71.
Again di and da lying respectively in the semi-latent regions of the first
species corresponding respectively to 71 and to 7a are normal to each other.
For (di|<^) = (di|(7ada + XaC2)) = 79(d,!d2), and (dg |<^) = 7i(da|d,).
Thus (di I </>da) = (da I ffdy) gives (71 - 7a) (di ] da) = 0 ; and hence (di | dg) = 0.
Similarly if /i be another point in the semi-latent region of the second
species of the root 71, such that <ft/i = 71/1 +Midi, then the same proof shows
that/i is normal to Ca, d, and /a; and so on.
Hence the semi-latent regions of different roots are mutually normal.
264 MATRICES. [chap. VI.
(6) Again consider the equation
{<h\<t>di) = (di\il>Ci).
This becomes 71 (Ci | di) + Xi (ci | Ci) = 71 (Ci | di).
Hence \(oi\oi)= 0.
Thus if Ci does not lie on the self-normal quadric, Xi = 0.
Now suppose that the latent region defined by Ci, c/, Ci"... does not touch
the self-normal quadric. Then it is always possible in an infinite number of
ways to choose Ci, c^', c/'... to be mutually normal and none of them self-
normal. Also the most general form for di is such that
fl}di = 7id!i + XjCi + Xj'ci' + . . ..
Then (ci |</>di) = 71 (Ci |di) + \i(c, \ci) = (di |</>Ci) = 71 (c, |di).
Hence Xi = 0, similarly Xi' = 0, Xi" = 0, and so on. Hence dj lies in the
latent region, and no semi-latent regions of the first or higher species exist
corresponding to the root 71.
(7) It is a well-known proposition that the roots of the equation
Ola, CLaa "" P> ••• ^
= 0,
are all real ; provided that ttpa = 0^/^, where all the quantities ttpa sure real.
Hence it follows that the latent regions of symmetrical matrices are all
real. For if 71 be one of the real roots, the equation <f>x = yiX, determines
the ratios of the co-ordinates of x by real linear equations. If the self-normal
quadric be imaginary owing to all the normal intensities being real [cf.
§ 110 (3)], a latent region, being real, cannot touch it. Hence in this case
there can be no latent self-normal point, such that ^p=:7pCp. Hence from
above there are no semi-latent regions. Thus finally in this case a complete
real normal system of the type Ci, Ci', Ci"... (?», Cj'..., C3..., c^, c/, c/' ... can be
found defining the latent regions of 71, 72, etc. ; each element being at unit
normal intensity.
(8) If the latent region defined by Ci, Ci', c/'... touches the self-normal
quadric (assumed real), but is not part of a generating region, take Ci to be
the point of contact, and take c/, Ci" ... to be mutually normal elements on
the tangent plane at Ci, but not self-normal [cf. § 113 (5)].
Then the general form of di is such that, <l>di = yidi + XiCi + X/ci' -I- . . . .
Hence {c, \ (fd,) = 71 (c^ | d^) = (d^ | <f>(h) = 7^ (cj \ di).
Also (0|';M) = 7i(Ci'|di) + X,'(Ci'|O = (dilK) = 7i(Ci'|^).
Hence Xi' = 0, similarly Xi" = 0, and so on.
Thus di is such that 0di = 7id!i -f- XiCi.
154] SYMMETBICAL MATRICES. 265
There can only be one independent point di satisfying this equation. For
if di' be another point such that (f>di' = yidi' + Vci, then it was proved in
§149 (4) that if Ci,di,di' are independent, then XiCi is independent of Xi'cj,
whereas here they are the same point ; which is impossible.
(9) If the latent region of the root 71 contain a real generating region of
p — 1 dimensions of the self-normal quadric^ let the points CmOuf'Cip ^^
chosen to be mutually normal points in this generating region [cf. ^ 79 and
80], and let the remaining points of the latent region be mutually normal
and normal to Cu ••• Cip» but not self-normal. Let these remaining points be
Let di be any point in the semi-latent region of the first species, but not
in the latent region.
Then <^ = jidi + 2 X/)y^ + ^Ci + fjLi'Ci' + . . ..
it=i
Hence (c, \(f>di) = 7, (Ci |di) + /^ (c, |ci) = (di | ifxh) = 71 (ci \di).
Hence fh = 0. Similarly /ai' = 0, /ii" = 0, and so on.
Hence <f>di = yidi + 2 \Oik'
Thus in the semi-latent region of the first species the subregion of highest
dimensions not necessarily intersecting the latent region cannot be of higher
dimensions than the real generating region contained in the latent region
[cf. § 149 (4)]. Similarly for the semi-latent regions of higher species.
164 Symmetrical Matrices and Supplements. (1) A one to one
correspondence of points to planes is given by the operation which transforms
the reference elements Ci, «» ... e„ into the planes Ai, -4a ... A,, where [cf. § 97
Prop. IV.]
and so on.
Then the element x (= Sfe) is transformed into the plane X(= X^A).
Now let 6i, 6a ... &y be a normal system, so that Ei = \ei, and so on ; then
^i = |(aiiei-haae2+...) = |ai, say.
Similarly A^ = Kaja^i -f- eugAi +••.) = I^> ^^^ so on.
Also let <!> denote the matrix -*- ^ — " .
Then the type of one to one correspondence of points to planes, which
we have been considering, can be denoted by X = \(f>x.
Similarly this type of correspondence could be denoted by <l>\x; but
1^ and ^1 are in general different operations.
(2) If to every point there corresponds a plane and to every plane there
corresponds a point, then the matrix ^ has no vacuity. In this case "
CLi f CEa • • • Cbp
266 MATRICES. [chap. VI.
is a determinate matrix; denote it by (f>'~\ Then if X= |^, |X = ^, and
In general the transformations!^ and <^~^| are different: thus \if>x is
different from ^~^ \x,
(3) If the latent roots of <f> are all unequal, then the operations |<^ and
^~*| can only be identical when the v latent points of ^ form a normal
system, that is, when the matrix is symmetrical ; and when, in addition, the
product of the latent roots of the matrix is unity.
For let Ci, Cg ••• Cr be the latent points of ^, so that ^ can be written
— — ^ _ ^
C\ , Cq • • • Cy
Then |^Ci=7i|c,. Hence |^Ci=^~^|ci, becomes yi\ci = (f>''^\ci, that is
7i<^|ci = |ci.
Assume that | Ci = XjCi + Xfi^ + . . . + \C^.
Then yi(f> | c, = yi\<l>Gi + yiX^C^ + . . . + y{Ky<\>G^.
But by § 141 (1) <l>Cj = y.//3 ... 7„Ci, <^02= 7,7s ... y^C^, etc.
Hence 71^ | Ci = 717a . . . yv\Ci -h 71^3 • • • y^^Gi + . . . + 71^2 • • • yp-i^t^G^
Hence since 71, 7a. ..7,, are all unequal, 717^ ... 7„=1, \2 = 0, \8==0,...\y=0.
Thus |ci = XiCi; and similarly for [cg, |cs, etc. Accordingly the latent
points of the matrix form a normal system, and the product of the latent roots
is unity.
(4) Conversely if the matrix be a symmetrical matrix with unequal
latent roots of which the product is unity, then |^ and <^"~*| are the same
operations.
For let Ci, Ca ... c„ be the latent points, 71, 7a ... 71. the latent roots of <f>.
Then Cj, Ca ... c„ are the latent points and 71"^ 72"* ... 7,,'* are the latent
roots of (fr\
Also |^Ci = 7i |ci, and <^|<^Ci = 7i<^|ci = 7i72... 7„|ci = |ci.
Hence |<^i = <f>~^ |Ci. Similarly for the other latent points.
Thus finally |<^ = ^~* \x,
(5) It is obvious that in this case the operation |^ is equivalent to the
operation of taking the supplements with respect to some quadric with
respect to which Ci, Cg ... c^ form a self-normal system. Let I denote this
operation; let 61, €3 ... €„ be the normal intensities of Ci, Ca ... c„ with respect
to this operation; and let Si, S^.^.B,, denote the normal intensities of
Ci, Ca ... c„ with respect to the operation |. Also put
A = 6162 . . . 6y , A = O1O2 • . • o^.
165] SYMMETRICAL MATRICES AND SUPPLEMENTS. 267
A A'
'YlA
But Ici = 1^ = 7i |Ci = ^— CgC, ... c„.
Thus 7i = ~i T" • Similarly 7a = -^ -r- , and so on.
A'*
Hence 7172 ... 7^ = 1 = . ^, , therefore A = A'.
Hence 7i = 4, 7a= 4, ... 7. = rk-
Thus the symmetrical matrix ^, with imequal roots of product unity, has
been expressed in the form |I; so that (f>x = \Ix.
The latent points of the matrix are the one common system of self-
normal points of the two self-normal quadrics corresponding to | and I;
and the relations between the latent roots and normal intensities are given
above.
166. Skew Matrices. (1) The matrix — — - -(=(f>) has important
properties in the special case when
Oi = ♦ + 02162 + Oji^ + . . . + a^e,,,
etc.,
where Oi2 + «ai = 0i •••> ®p«r+ o^p^O ... , and «i, eg... 6„form a normal system at
unit normal intensities. Let such a matrix be called a skew matrix.
Then (Cp |^p) = 0, (e^ \<f>e^) = a.rp = - ap<r = - («p l^^e^).
Thus (x \<j>x) = 0 (A), and (x \<l>y) + {y |^) = 0 (B), whatever points x and
y may be.
(2) Any latent point Ci of this matrix, such that ^i = 7iCi, where 7, is
not zero, is self-normal. For from equation (A) (cj | <^Ci) = 7j (c^ | Cj) = 0.
Again, putting Ci and c, for x and y in equation (B), where c, is another
latent point,
(7i + 72)(ci|c9) = 0.
Hence either 71 -h 7, = 0, or (ci |ca) = 0.
(3) Assume that there are no repeated roots. The self-normal quadric
contains generating regions of dimensions ^ — 1 or — 1, according as v
be even or odd (cf. § 79).
If 1/ be even, ^ mutually normal elements Ji,j2> "*> can be found on the
quadric, defining one generating region, and ki,kt ... K mutually normal
i
268 MATRICES. [chap. VI.
elements defining another generating region. Also any element such as j^
can be made normal to all the k'&, except ip, and conversely k^ is normal to
all the/s, except jp (cf. § 80).
Then ji, ja •••>» ki, k^ ...ky can be chosen as the latent points of the
8 8
matrix. If 71, 72 ••• 71^ be the latent roots corresponding to juj^ ••.>» then
8 8
by subsection (2) —71, —72 ... —7^ are the latent roots corresponding to
8
K\ y AmjJ . • • tCy «
8
Hence if a? = f 1 ji + f 2 ja + . . . + f ^jV + i/A + 172^:2 + . . . + fi^K,
S 8 8 8
then ^ = 7]f 1 ji + 79^2^2 + . • • - liViki - 72^2^2 - • • • •
Thus (a; |<^) = (7,^1171 - 7ifi^i) O'l |*?i) + • .. = 0.
(4) K 1/ be odd, mutually normal elements of the type jf, can be
1
found, and — ^ — of the type &p, and an element e, not on the quadric, normal
to all the/s and all the A;'s.
Let these be the latent points of the matrix, then the element e must be
a null point of the matrix.
If a: = fe 4- %^j -h %7iky then ^a? = 27^; — ^^rik ; and (a? | ^) = 0.
(6) Assume that there are repeated roots. Let the roots 71 and 7, be
both repeated, and neither zero. Let di and da be in the semi-latent regions
of the first species (and not in the latent regions) corresponding to 71 and 7a
respectively, /, and f^ in the semi-latent regions of the second species (and
not in the semi-latent regions of the first species), and so on. Let Ci, Ci', ...
be in the latent region of 71, and Ca, Ca^ ... in that of 7,.
Then we may assume [cC § 150 (5)],
^1 = 7iCi, <A^i = 7 A + XiCi, <^/i = 7,/i -f- /tiidi, and so on.
Hence by equation (B), (ci \4>Oi) + (c/ |<^Ci) = 271 (ci |c/) = 0.
Hence Ci and c/ are mutually normal as well as being self-normal. Thus
the latent region of a repeated root is a subregion of some generating region
of the self-normal quadric.
(6) Again from equation (B), (ci' \if>d^) -f (cZj |<^c/) = 271 (c/ |di) = 0,
Hence the semi-latent region of the first species corresponding to 71 is
normal to the latent region corresponding to 71.
Also by equation (A), (di |<^di) = 71 (dj |di) = 0.
Hence (di |di) = 0. Therefore each point in the semi-latent region of the
first species is self-normaL Further if d/ be another point in this semi-
latent region,
{d, \<f>d,') -h (d/ I H) = 271 (di |d/) = 0.
155] SKEW MATRICES. 269
Thus (di|d/) = 0. Hence the semi-latent region of the first species is a
subregion of a generating region of the self-normal quadric ; and therefore
the latent region and semi-latent region of the first species are together
contained in the same generating region.
(7) The same proof applies to semi-latent regions of higher species.
Hence the complete semi-latent region (which contains the latent region)
corresponding to a repeated root is a subregion of a generating region of the
self-normal quadric.
(8) The same proof shows that the complete semi-latent region of one
repeated root 71 is normal to the complete semi-latent re^on of another
repeated root 7^ unless 71 + 72 = 0.
(9) Again assume that the matrix is of vacuity a and of nullity 0.
Let c be any null point of the matrix, and Ci any latent point corresponding
to the non- vanishing root 71.
Then ^ = 0, hence (ci |^) = 0. Thus by equation (B)
(c, \4>c) -h (c |</>Ci) = 7i (c |c) = 0.
Hence the null region is normal to the latent regions of all the other latent
roots.
Similarly the null region can be proved to be normal to all the semi-
latent regions of the other latent roots.
(10) Let (2 be a point in the vacuous region of the first species : assume
<l>d = Xc, where c is a null point.
Then (c\<f>d)-\-(d\(f>c) = \(c\c) = 0, by equation (B).
Hence either \ = 0, and c2 is in the null space; or (c|c) = 0, that is to
say, c is self-normal. Hence the subregion of the null region associated
with the vacuous region of the first species is self-normal.
Also from equation (A), (d \<l>d) = \{c\d) = 0,
Hence, assuming that X is not zero, (c \d) = 0, that is to say, d is normal
to c.
Again let & be any other null point, then (c' \<f>d) + (d |^') = X (c |c') = 0.
Hence, assuming X:^^, c is normal to every other null point.
(11) Similar theorems apply to vacuous regions of higher specie&
/
BOOK V.
EXTENSIVE MANIFOLDS OF THREE DIMENSIONS.
CHAPTEK I.
Systems of Forces.
166. Non-metrical Theort of Forces. (I) The general theory of
exteDsive manifolds, apart from the additional specification of the Theory of
Metrics, has received very little attention. It is proposed here to investigate
the properties of Extensive Manifolds of three dimensions, thereby on the one
hand illustrating the development of one type of formuIsB of the Calculus of
Extension, and on the other hand discussing properties which are important
from their connection with Qeometry*.
(2) Since in this case four independent points define the complete
region the simple extensive magnitudes are only of three orders, the point,
the linear element, the planar element. Also the only complex extensive
magnitudes are systems of linear elements. A linear element, — ^in that (a) it
is an intensity associated with a straight line, (J3) it is directed along the line,
so as to be capable of two opposite senses, (7) it is to be combined with
other linear elements on the same line by a mere addition of the intensities
[cf. § 95 (1)], — has so far identical properties with a force acting on a rigid
body. Only in an extensive manifold no metrical ideas with respect to
distance have been introduced. The other properties of a linear element,
whereby it is defined by two points and is combined with other linear
elements on other lines form a generalization of the properties of a force so
as to avoid the introduction of any notion of distance. It will be noticed
that the theorem respecting the combination of Forces known as Leibnitz's
theorem expresses the aspect of the properties of forces which are here
generalized. The parallelogram of forces is without meaning at this stage
of our investigations: for the idea of a parallelogram depends on the
Euclidean (or equivalent) axioms concerning parallel lines, and such axioms
presuppose metrical conceptions with respect to distance which have not yet
been enunciated.
* The formiila and proofs of propositions in this hook are, I heliere, new. Many of the
propositions are well-known ; bnt I belieye that they have hitherto been obtained in connection
with Metrical Geometry, either Eadidean or non-EacUdean.
w, 18
274
SYSTEMS OF FORCES.
[chap. I.
(3) We shall therefore use the term force as equivalent to linear element,
meaning by it the generalized conception here developed apart from metrical
considerations. It will be found that very few of the geometrical properties
of ordinary mechanical forces are lost by this generalization.
Also, when no confusion will arise, plane will be used for planar element.
The context will always shew the exact meaning of the term.
167. Recapitulation of Formuls. (1) It will be useful to re-
capitulate the leading formulas of the Calculus of Extension in the shape in
which they appear, when the complete manifold is of three dimensions.
(2) The product of four points is merely numerical The product of a
linear element and planar element is the point of intersection of the line and
plane. The product of two planar elements is a linear element in the line of
intersection of the two planes. Thus a linear element can be conceived
either as the product of two points or as the product of two planar elements.
The product of three planar elements is a point The product of three points
a planar element. The product of a linear element and a point is a planar
element. The product of two linear elements is merely numerical.
(3) The formulae for regressive multiplication are [cf. § 103 (3) and (4)]
dbc ,de=de. obc — (abce) d — {abed) e = ( abde) c + (cade) b + (bode) a. . . (1 ).
Thus five points a, 6, c, d, e are connected by the equation
(bcde) a - (acde) b + (abde) c — (ahce) d + (ahcd) e = 0 (2).
Again abc . def= (abcf^de + (abed) ef+ (abce)fd
= (adef) be + (bdef) ea + (cdef) ab — -- def. abe (3).
By taking supplements, we deduce that these formulae still hold when
planar elements A,B,C, D, E, F are substituted for the points a, 6, c, d, e,f,
(4) Also from § 105 there come the group of formulae, B„ -B,, JB,, P4 being
planar elements,
(a,a,.5A) = (a,50(«*B.)-(«iA)(a.Bi) ^W;
(a,a^.B,BA)^ ((hB^\ ((hB;)A<hBt\
(a^,\ (o^,), (o^,),
(a^,\ (afi,\ (a^,\
(oiChWii) (JB,JB^A) = (oiOiichfii . B^BJBA) =
(a,B,), (a,B,l (a,B,), (a,B,X
(a^,\ (o^.), (o^,), (a^,\
(a^,\ (a»B^, (0,5,), (0,5,),
(aA), (aA), (aA), (a,B,\
(5);
.(6).
157, 168]
RECAPITULATION OF FORMULAE.
275
(5) Also &om equation (4) a useful formula may be deduced by putting
Bi = bcci, JB, = bcc^. Then from equation (4)
OiO, . (fccci) (bcCi) = (oi&cci) (ajbccj) — (oibcc*) (ajbccj).
But from § 102, (hcCi) (bcCi) = (bcCiC^) be.
Therefore OiO, . (bcCi) (ftcca) = ((h<^o) (ftcCiCj).
Hence finally, (<h<^) {bcCiOs) = ((hbcci) {(Q>cc^ — (^i^Cj) (oJkc^ (7).
This equation can be written in another form by putting F for the force
be. Then
{oxoJF) (cc^) = (a,<kF) (a^)-((hC,F) (a^F) (7').
168. Inner Multiplication. (1) If a be any point, then |a is a
planar element; and if J. be any planar element, then {J. is a point. If F be
a simple linear element, then |^ is a simple linear element ; and if iSf be a
system of linear elements, then |iS is a system of linear elements.
(2) Again[c£§99(7)]. ||a = -a,|U=-^||l'=^
Also (cf. § 118), \(al)c\de)^\de.\abc =(cfo|a6c), |
\(de \ahc) = ( (abc \de) = - (abc\de))
and hence
Also
and hence
Also
hence
Finally
.(8).
.(9).
|(a6c |d) == — \ahc .d^{d |a6c),)
|(d laic) = ||(a6c |d) = (aftc lei) j
\{ab |c) = — |a6 . c = - {c\ah) ;]
.(10).
.(11).
.(12).
|(c|a6) = -||(a6|c) = (a6|c)
(a |6)== (6 |a), and (at |cd)= (cd |a6),)
and {abc \def) == {def \ abc) J
(3) Again from the extended rule of the middle factor (cf. § 119),
abc \de = (ab \de) c + (6c \d€) a + (ca \de)b.
And de \abc = | (abc \de)-lde\bc)\a-\- (de \ca) \b + (ab \de) | c\
Again oic |d = (a|d)6c + (6|d)ca + (c|d) a6.
And d \abc = \(abc |d)= (d \a) \bc + (d |6) \cd + (d \c)\ab
Again a6|c = (a|c)6 — (6|c)a. )
And c|a6 = -|(a6|c) = (c|6)|a-(c|a)|6j
(4) Againfrom§120,(a6|cd) = (a|c)(6|d)-(a|d)(6|c) (16),
(17),
.(13).
(14).
.(15).
(abc\d€f) = \ (aid), (a\el (a\f)
(6ld), (6|6), (6|/)
(c|d), (c\el (c\f)
(a\eh {a\f\ {a\g\ (ajA),
{h\e\ {b\fl{b\g\{b\h\
(c \e\ (c \f\ (c 1(7), (c |A),
(dk), (d|/), (d|fl^), (^jAX
{abcd\efgh)-
(18).
18—3
276 SYSTEMS OF FORCES. [CHAP. I.
(5) It is unnecessary to reproduce the special forms of the more general
but less useful formulae in § 122. These eighteen formulse of the present and
the preceding articles are the fundamental formulae which will be appealed to
as known. They are all immediate consequences either of the extended rule
of the middle factor or of the formula of § 105.
169. Elementary Properties of a Single Force. (1) A force can
be represented as a product of any two points in its line. This is a simple
corollary of § 95.
(2) A system of forces lying in one plane is equivalent to a single force.
This is a corollary of § 97, Prop. IV.
(3) A force can be resolved into the sum of two forces on lines concurrent
with it and coplanar with it. For let a be the point of concurrence, then db
can be chosen to represent the given force. Two' points b and d can be found
on the other lines respectively, such that 6 ^ Xc + fjtd. Hence ab = \cu) + /lad.
Thus a6 is resolved as required.
(4) Any force can be resolved into the sum of two forces, of which one
passes through a given point and one lies in a given plane, which does not
contain the point.
For consider the plane P through the given force and the given point.
It cuts the given plane in a line concurrent with the force, and through the
point of concurrence a line can be drawn in P through the given point : then
two forces can be found by (3) along these lines of which the sum is equivalent
to the given force.
Thus if a be any given point, A any given plane, F any given force; then
we can write,
F=:ap+AP.
160. Elementary Properties of Systems of Forces. (1) The
letter S will only be used to denote a system of forces. Two congruent
systems of forces (i.e. of the types S and \8) will be spoken of as the
same system at different intensities. If ^i, F2, etc. be any number of
forces, then S = XF represents the most general type of system.
(2) If a be any given point and A any given planar element not
containing a, any system of forces (8) can be written
i8f = op + ^P.
For by § 159 (4), F, = ap, + AP,, J^a = ap^ + AP^, etc.
Hence fif = ^1 + i; + ... = a(^i + pa + ...) + il (Pi + P^ + ...) = op + AP.
Hence any system can always be represented by two forces of which one
lies in a given plane, and one passes through a given point not lying in the
plane.
159 — 162] ELEMENTARY PROPERTIES OF SYSTEMS OF FORCES. 277
(3) The mention ofp and P can be avoided by means of the formula
{aA)8 = a.A8 + aS.A.
This can be proved as follows. From (2) of this article
S^ap + AP.
Multiplying by a, we have aiS = a . -4P = (aP) A — (ail) P.
Hence aS.A^- (aA) PA = (aA) AP.
Again multiplying by -4, we have AS = A .ap — (Ap) a - (Aa)p, ^
Hence / a . A8 = (aA) ap.
The required formula follows at once.
(4) It follows from (2) that any system 8 can be expressed in the form
8^ab + cd.
For we may write cd instead of ilP in the expression for 5. It will be
proved in § 162 (2) that one of the two lines, say ab, can be assumed
arbitrarily.
(5) If Ci, e^y e^, €4 be any four independent elements, then [cf. § 96 (1)]
8 can be written
Hence any system can be represented as six forces along the edges of
any given tetrahedron.
When Ci, e^, ^, e^ are unit reference elements, tti,, etc. will be called the
co-ordinates of the system 8.
161. CoNDniON FOR A SiNOLE FoRCE. (1) If /S be any system of
forces, {88) is not in general zero. For by § 160 (4), 8 may be written
ab-\'Cd) hence {88) = 2 {ahcd).
Thus {88) only vanishes when {abed) = 0, ie. when ah and cd intersect.
But in this case ah-\-cd can be combined into a single force.
Thus {88) =: 0, is the required condition that 8 may reduce to a single
force.
(2) If i8f = op + ilP, then
{88) = 2 (op . AP) = 2 {aA){pP) - 2 {aP) {pA).
then J {^8) = ttuTTi* H- ttjiTTm + ir^nr^.
(3) If 8 reduce to a single force, \8 reduces to a single force. For if
{88) = 0, then \{88) = 0, that is (|i8f \8) = 0.
162. CJoNJUQATE Lines. (1) When a system 8 is reduced to the sum
of two forces a& and cdy then the lines ab and cd are called conjugate lines,
and the forces ab and cd are called conjugate forces with respect to the
system. Also ah will be called conjugate to cd, and vice versa.
278 SYSTEMS OF FORCES. [CHAP. I.
(2) To prove that in general any line ab has one and only one conjugate
with respect to any system S, not a single force.
For if 8 = Xa6 + fAcd, then 8 — Xa6 is a single force.
Hence {(8 - Xa6) (8 - \ab)} = 0 ; that is (88) - 2\ (ab8) = 0.
Therefore X = q , ,^^v ; and hence /S— ^ . h^\^ represents the force
conjugate to Xab. Since only one value of X has been found, there is only
one such force; and if (ai8) be not zero, there is always one such force.
Similarly if any line be symbolized by AB, its conjugate with respect to 8
'^^~2(ZaS)'*^-
(3) If two lines ah and od intersect, their conjugates with respect to
any system 8 intersect.
For by multiplication
r 2 {ah8) ""^J r 2 (pd8) "^J ^ 4 {oh8) (cd8)
since by hypothesis (ahcd) = 0.
= 0,
163. Null Lines, Planes and Points. (1) If L be any force, and
(L8) = 0, then the line L is called a null line of the 8yst>em 8.
Note that L can be written in the two forms db and AB ; the product
{ab8) is a pure progressive product ; the product (AB8) is a pure regressive
product.
If F be any force, then (F8) is called the moment of 8 about the force F.
(2) The assemblage of null lines of any given sjrstem 8 will be called
the linear complex* defined by the system 8,
(3) If a be any point, then the planar element a8 defines a plane
containing a, which is called the null plane of the point a with respect to
the system 8.
If A be any plane, then the point A8 lies in A and is called the null
point of the plane A with respect to the system 8.
* Linear Complexes were first inyented and stadied by Pluoker, cf. PhU. Tram. toI. 155, 1805,
and his book New Oeometrie des Raumes, 1868. The theory of Linear Complexes is developed in
Clebsoh and Lindemann's Vorlenmgen Uher Oeometrie^ toL 2, 1891 ; also (among other plaoes)
in Koanig's La QSamitne ReglSe, Paris, 1896, and in Dr Rudolf Sturm's LirUengeometrie, 8 vols.,
Leipzig, 1892, 1898, 1896. The chief advanoes in Line Geometry, since Pliioker, are due to
Klein. Buchheim first pointed out the possibility of applying Grassmann*s Auidekfutngslekre to
the investigation of the Linear Complex, cf. On the Theory of Screws in Elliptic Space, Proe, of
London Math. Soc. vols, xv, xvi, and xvn, 1884 and 1886.
163, 164] NULL LINES, PLANES AND POINTS. 279
164 Properties of Null Lines. (1) All the null lines of 8 which
pass through any point a lie in the null plane of a ; and conversely all the
null lines which lie in any plane A pass through its null point. For if ah
be any null line of 8 through a, then (abS) = 0 = (a8 . b). Hence 6 lies on
the plane a£f.
Similarly if AB be any null line of 8 in A, then {ABS) = 0 = (A8 .B).
Hence B contains the point AS.
(2) If a lie on the null plane of 6, then 6 lies on the null plane of a.
For (6S.a) = 0 = -(a/Sf.6).
It is obvious that in this case ab is a null line.
(3) If any null line £ of a system of forces intersect any line ob, it
intersects its conjugate.
For by hypothesis, {8L) = 0 = (obL).
««°<* ^(^-§(S«*)=«-
(abS)
Also obviously any line intersecting each of two conjugates is a null line.
(4) The conjugates of all lines through a given point a lie in the null
plane of a.
For let ab be any line through a. Then the plane through a and the
conjugate of at is defined hy a\S — ^ / Tq\ ^l > *^** ^' ^y ^'
It follows as a corollary that a8 . b8 represents the line conjugate to ab.
For this conjugate lies in the line of intersection of the null planes of a and
6. Thus
aS.bS = 8-l^^ab.
2 {abS)
(5) If the system do not reduce to a single force, no two points have
the same null plane and no two planes have the same null point
For if X and y be two points such that x8 = y8, then putting x^y + z,
zS s* 0. Hence by § 97, Prop. L, S = g) ; and therefore 8 reduces to a single
force, contrary to the assumption. Thus no two points with the same null
plane exist.
If X and F be two planes with the same null point, then X8 = YS.
Hence by taking supplements jJT |£i = |F|iSf. But |iS is a system of forces,
and hence the points \X and \Y cannot have the same null planes with
regard to it unless \8 reduce to a single force. Hence from § 161 (3) X
and Y cannot have the same null points with regard to 8, unless 8 reduce
to a single force.
280 SYSTEMS OF FORCES. [CHAP. I.
(6) The relations between planes and their nail points and between
points and their null planes can be expressed in terms of ordinary algebraic
equations involving their coordinates*. For let
and X = Xi^f^4 — ^^^^4 + ^6a«4 — ^4^«A-
Then the equation, either of a plane through x, or of a point on X, is
{xX) = (Xif 1 + X,ft + X^f , H- X4?4) {e,e^,) = 0.
Also [cf. § 160 (5)] let S be the system
ttii^iea + Ou^A + aif«A + ^afiifi^ + ai4^«4 + ^hfi^*
Then by simple multiplication x8 = (a^i — ttu^a + otis^,) ^i^s^ + etc.
Hence the co-ordinates Xi, X,, X,, \ of the null plane of a; can be written,
<rXi= • + 034fs + a4ifi + a«f4,
<rX, = a4»fi+ • +ax4?» + «n?4,
o-X, = o^f 1 + «4if a H- * H-ttuft,
<rX4 = o«f , + ai,f J + Onf , + * ,
where we assume a^ H- On = 0 = oti, + On = etc.
Again by simple multiplication, we find
XS = ( » + OnX, + fl^X, + 041X4) (^^1^4) ei + etc.
Hence the co-ordinates fi, fs, f„ ^4 of the nujl point of X are given by
o" f 1 = * + O11X3 H- ottiXs H- 0141X4,
</fa = ai,Xi+ ♦ +a«X,-|-a4,X4,
0"'f I 5= flisXi + OgsXa H- ♦ +€(43X4,
O" f 4 = OmXi + 094X3 + OS4X9 -h * .
(7) Thus, if the reference elements be normal points at unit normal
intensities, a skew matrix [cf. § 155] in a complete region of*three dimensions
operating on x can be symbolized by \x8.
166. Lines in Involution. (1) A system of forces can always be
found so that five given lines are null lines with respect to it. But if six lines
are null lines with respect to some system, their co-ordinates must satisfy
a condition.
For let Li, £,, £,, L4, L^, L^ be any six independent lines. Then
[cf. § 96 (2)] we may write any system /S, ,
Assume that {L^S) = 0 *= {L^ = {L^ = {LS) = (/iaS).
* Cf. Glebsch and Lindemann, VorUnmgen ilber Geometriet vol. 11. pp. At et Beq.
165, 166] LINES IN INVOLUTION. 281
Then the five ratios f i : f t : f j : f i : f 6 : ft are determined by the five
equations
ft(iA)+ ♦ +6(/iA)+ft(iiA) + ft(i2£5) + ft(iJi«)=o,
ft {LJ^) + ft (i^,) + ft {L,L,) + ft (i^,) + ♦ + ft {LJ..) = 0.
Hence 8 is completely determined. Therefore one and only one system
of forces can in general be found such that the five lines Z^, £,...£5 are null
lines with respect to it.
(2) If L^ be also a null line with respect to the same system then
eliminating ft, ft, etc. from the six equations of condition, we find
♦, (AA), (Ai,), ... (Aia)
= 0;
{Ld.,\ (L,L,l , ♦
where it is to be noticed that (£i£s) = {LJL^,
(3) Definition. Six lines which are null lines with respect to the same
system are said to be in involution ; and each is said to be in involution with
respect to the other five.
Thus the propositions of the preceding article can be stated thus :
The lines through a given point in involution with five given lines lie in
a plane, cf. § 164 (1).
The lines in a given plane in involution with five given lines are con-
current, c£ § 164 (1).
Again, a linear complex may be conceived as defined by five independent
lines belonging to it.
166. Reciprocal Systems. (1) Two systems of forces 8 and 8' are
said to be reciprocal* if {8 ST) = 0.
It is obvious that a force on a null line of any system is a force reciprocal
to the system.
(2) If two systems be reciprocal, the null lines of one system taken in
pairs are conjugates with respect to the other system.
For let 8 and fif be the two systems. Then (8S) = 0. Let a& be a null
line of 5, its conjugate with respect io Sf \a S — ^ ) , ^ ah.
* Beciprooal systenn of meohanioal forces were first studied by Sir B. S. Ball, of. Transactions
of the Royal Irish Academy, 1871 and 1874, vol. 25, and Phil. Trans. (London), voL 164, 1874,
and his book Theory of Screws (1876), oh. ni. The theory of systems of forces for non-Euolidean
Geometry was first worked out by Llndemann in his classical memoir, Mechanik bei Prcjectiven
Maasbestimmung, Math. Annal. voL vii, 1873. The most complete presentment of Sir B. S. Ball's
Theory of Screws is given by H. Gravelias, Theoretische Mechanik, Berlin, 1889.
j (19).
282 STSTEHS OF FOBCES. [CHAP. I.
Hence the conjugate of ab with respect to ST is a null line of S.
It is to be noted that there are conjugates of either system which are not
null lines of the other.
167. FoRMULfi FOR Systems of Forces. (1) The following formulae
are obvious extensions of the standard formulae of § 157, remembering the
distributive law of multiplication.
From equation (I), § 157,
(ibc.S==S. abc =^ (abS) c + (caS) b -\- (bcS) a.]
Also 8c.de — (See) d — iScd^e
From equation (3), abc.d8=^ (adS) be + (bdS) ca + (cdS) ah (20).
By taking supplements, and replacing \S hy 8, we see that the formulae
hold when planar elements replace the points.
(2) To prove that, if a be any point and 8 any system of forces
8. ai8f= a8.8^i(SS)a,
8.A8^A8.8^HS8)A
For let 8=^bc + de.
Then 8.aS=de.(ibc + bc.ade-- (abed) e + (ahee)d - (ahde) e H- (occfo) 6
= (bcde) a ; from § 157, equation (2).
Also (88) = 2 Q)cde). Hence 8.a8^\ (88) a.
The second formula follows by taking supplements.
(8) To prove that
a8 . 65= (ab8) S-J (iSfflf)a6, )
A8,B8^(AB8)8''^(88)Ab] ^^^'•
For let /S = Xa6 + erf. Then a8.b8^acd.bcd^(ahed)cd.
But (oftcd) = (oi/S), and «i = ^-ii^^«*-
Hence a8.b8^(ab8) 8-^(88)01.
This forms another proof of the corollary to § 164 (4).
(4) From equations (21) and (22) it is easily proved that
a8 .b8 .e8:=i(88){(bc8)a + (eaS)b + (abS)e}=-^(88)8,abe]\ .« .
A8.B8.C8-^^(88)8.ABC |...(.^.i).
Also from equation (22), aS.b8,8 = ^ (88) (abS), \ .^ .
A8.B8.8=^(8S)(AB8)] ^^^•
} (21).
.. _i
167] FORMULiE FOR STSTEMS OF FORCES. 283
(5) To prove that if a be any point and S and 8' any two systems of
forces, then
S.aS'+8\aS = (8Sf)a,
8.A8'
+ 8\a8 = (88')a,\ .^..
+ 8'.A8=^(S8')A] ^ ^'
For in equations (21) write 8+8' instead of 8.
Then (8 + 8').a(8 + 8')^^{{8-\-8r}(8-\-8')}a.
Hence by multiplying out both sides,
8.aS + ff.a8' + 8.a8' + S'.a8^^{88)a + ^(8'8')a-\-(88')a.
But 8. a8^i (88) a, and iST . ofif' = J (S'fiT) a. Hence the required result.
Similarly from equation (22) we can prove
a8. b8' + a/Sf' . b8 = (ab8) 8' + (ab8') 8^(88') ah, \
A8.BS: + A8r.B8^{AB8)Sr^-{AB8r)8^{8Sr^AB] ^^^^'
CHAPTER 11.
Groups of Systems of Forces.
168. Specifications of a Group. (1) I{ SuS^, ,,,S^^ he any six
independent [of. § 96 (2)] systems of forces, then any system can be written
in the form X^Si + X^t + ... + X^e. Let Xi, X,, ... X« be called the co-ordinates
of ^ as referred to the six systems.
Definitions. The assemblage of systems, found from the expression
XiSi + \^i by giving the ratio X^ : X, all possible values, will be called a ' dual
group ' of systems. The assemblage of systems, found from the expression
XiSi + XjS, + X,S, by giving the ratios Xi : X, : Xs all possible values, will be
called a ' triple group ' of systems.
The assemblage, found from XiSi + \S^ + X,Sj + \4^^ by giving the ratios
X| : X, : X, : X4 all possible values, will be called a * quadruple group/ The
assemblage, found from \iSi + \^^-\-\^t'^\4^4 + \t^i by giving the ratios
Xi : X, : X, : X4 :X« all possible values, will be called a ' quintuple group.'
(2) A dual group will be said to be of one dimension, a triple group of
two dimensions, and so on.
It is obvious that a group of p — 1 dimensions (p = 2, 3, 4, 5) can be
defined by auy p independent systems belonging to it ; and also that not
more than p independent systems can be found belonging to it.
(3) Again, if the co-ordinates Xi, X,, ... X« of any system S satisfy a linear
equation of the form,
ffiXi + OsX, + a,X, + 04X4 + o^Xs + OjX^ = 0,
then S belongs to a given quintuple group.
For by eliminating X,, we can write
a.S = Xi (a^i - flTiSe) + X, (daSf, - 0,5, ) + X, (a ,Sf, - aJS,)
+ X4 (aeS4 - a4Se) + X, (0,55 - a^«).
Hence flre^i — ai^e* «^8 — «a'S»e, eta, define a quintuple group to which 8
belongs.
168, 169] SPECIFICATIONS OF A GROUP. 285
Similarly it can be proved that if the co-ordinates Xi...X« satisfy two
linear equations SaX = 0, Xl3X = 0, then the system must belong to a certain
quadruple group: if the co-ordinates satisfy three linear equations, the.
system must belong to a certain triple group : and if four linear equations, to
a certain dual group.
(4) Hence a dual group may be conceived as defined by two systems
belonging to it, or by four linear equations connecting the co-ordinates of
any system belonging to it.
And generaUy, a group of p — 1 dimensions (/> = 2, 3, 4, 5) is defined by p
independent systems belonging to it, or by 6 — /> linear equations connecting
the co-ordinates of any system belonging to it.
169. Systems Reciprocal to Groups. (1) Definition. A system of
forces, which is reciprocal to every system of a group, is said to be reciprocal
to the group.
If a system S' be reciprocal to p independent systems, Si, S^, ... S^o{ a
group of p — 1 dimensions, it is reciprocal to the group.
For any system of the group is £f = \8i + ... -I- X^p.
Hence (SS') = \ (S^S') + . . . -h Xp {8^').
But by hypothesis (S^S") = 0 = {SJS") = . . . = (Spflf'). Hence (SS') = 0.
(2) All the systems reciprocal to a given group of p — 1 dimensions form
a group of 5 — p dimensions.
For let ^1, £„ ... ^e be any six independent reference forces.
Then any system can be written
S = Xi^i + "KjS^ + . . . + Xft£^6.
If this system be reciprocal to the p independent systems Si, S^, ... S
which define the given group, then the following p equations hold :
X, {EiSi) + X, (E^i) + ... + X,(^,fif,) = 0,
Xi (EiS,)+\, (E^,) -h ... +MEA) = 0,
2»
ft
Xi(EiS,) + \,(E^,) + ...+MJSA)^0.
Hence by § 168 (4) the group of reciprocal systems is of (5— p) dimensions,
and is therefore defined by any (6 — p) independent systems belonging to it.
(3) Definition, Let this group of reciprocal systems be called the
group reciprocal to the given group; and let the two groups be called
reciprocal.
It is to be noted that there is only one S3r8tem reciprocal to a quintuple
group ; or in other words, the reciprocal group is of no dimensions.
286 GROUPS OF SYSTEMS OF FOBCES. [CHAP. II.
170. Common Null Lines and Director Forces. (1) Definition.
A line which is a null line of every system of a group is called a ' common
null line of the group.'
It is obvious that if a line be a null line of p independent systems of a
group of (p — 1) dimensions, it is a common null line of the group.
Definition, Those systems of forces of a group which are simple, that is,
which reduce to single forces, are called ' director forces of the group ' ; and
the lines, on which they lie, are called ' director lines of the group.'
(2) Since the null lines of a system are the lines of forces reciprocal to
the system, it follows that the common null lines of a group must be the
director lines of the reciprocal group ; and conversely.
(3) Let Si, flfj, ... Sp define a group of /> — 1 dimensions, and let fifp+i,
fifp+j, ... St define the reciprocal group.
Call the first group 0, the second group 0\
Then if \iSi + XaS,+ ... + Xpflfp be a director force of 0, we must have
(\8^+ ... + xls;) (XxSi + ... +XpSp) = 0.
Hence \* (S^S;) + 2XaX, (S^S;) + . . . + X^» (S^;) = 0.
Let this equation be called the director equation of the group O.
If ai:a,:...:ap be a system of values of the ratios XaiX^: ...:\ which
satisfy this equation,, then aiSi + aJS^ + . . . + a,^p is a director line of O and a
null line of 0'.
Similarly if Xp+i<yp+i+ ... + X^/ be a director line of 0', the X's must
satisfy the equation
\%+, (S^p+i iSTp+O + 2Xp+i\p+, (S'^.S'f,^;) + etc. = 0 ;
and the director line of 0' is a null line of 0,
(4) A common null line of the group G is a null line of any one of its
director forces i^. But the null lines of a single force are the lines inter-
secting it. Accordingly each common null line of a group intersects all the
director lines and conversely.
171. Quintuple Groups. (1) Let a quintuple group be defined by
the five systems Si, S^, S^, S^^ 8^ and let 89 be the system which forms the
reciprocal group.
The director equation is
V(fi^i'Si) + 2\X,(SA)+...+V(SA) = 0.
Ka^ 10^:0^:04:0^ satisfies this equation, then OiSi-k- 0^^+ Otffi-\- o^^^-i-oJS^
is a director line of the quintuple group ; and accordingly is a null line of 8^',
Hence the director lines of a quintuple group form a linear complex
defined by the system St [cf. § 163 (2)].
Thus conversely a linear complex may be said to be defined, not only by
170 — 172] QUINTUPLE GROUPS. 287
any five independent lines belonging to it [of. § 165 (3)], but also by any five
independent systems of the group reciprocal to fif/.
(2) Also if 6161^4 be the four co-ordinate points and any system S be
denoted by 'Wuei6i + 7rMV4 + 'Wi5ei6i + '7r^4ei + '7ri4^«4 + 'Tasej6s, then a linear
complex is defined by the two equations
2a7r = 0 (1),
and '"'n'Tu + w'i,7r4j + WmTTj, = 0 ( 2 ) ,
where the a's are given coefficients.
For the first equation secures that the variable system S belong to
a given quintuple group, and the second that it be a director force of the
group. Then by subsection (1) the lines, on which these director forces lie,
form a linear complex.
(3) The system reciprocal to the quintuple group given by equation (1)
can easily be expressed. For let this equation be written at length in the form,
Then the system, jS/ = aM^ei + a»4V4 + «u^^ + «4a«4^ + «i4^^4 + an«8^, is
reciprocal to any system S, whose co-ordinates satisfy equation (1). Therefore
89 is the required system. All the lines of the linear complex are null lines
of/g,'.
(4) In general a quintuple group has no common null line. But if the
reciprocal system reduce to a single force, then this line is the common null
line of the group. The linear complex is in this case called a special linear
complex. It consists of the assemblage of lines which intersect the line of
the reciprocal force.
172. Quadruple and Dual Groups. (1) Let S^ and S^ define a dual
group and 8^', 84, 8^^ 8^' the reciprocal quadruple group. Let the dual group
be called 0 and the quadruple group 0\
The director equation of (? is
V {S,8{) + 2X,\^ (8 A) + V {8^,) = 0.
This equation is a quadratic in Xi/X,, and has in general two roots, real or
imaginary. Let tti/a, and A/A be the roots, assumed unequal [cf. subsection
(9) below].
Then aiA + ^A and AA + iSA are the only two director forces of
the dual group 0.
Thus a dual group has in general two and only two director forces ; and
a quadruple group has two and only two common null lines.
Another statement of this proposition is that two systems of forces have
one and only one common pair of conjugate line&
(2) Also the common null lines of a dual group are the lines intersecting
the two director lines of the group ; and the director lines of a quadruple
group are the lines intersecting the two common null lines of the group.
288 QROUPS OF SYSTEMS OF FORCES. [CHAP. U.
(3) DefinUion. The assemblage of common null lines of a dual group is
called the ' congruence ' defined by the group.
Thus the lines of a congruence are lines intersecting two given lines.
The lines indicated by the director equation of the group 0\ namely
V(^/S,0 + 2\,\(S,'3:)+ ... + V WS.')«0,
form the congruence defined by the group Q.
(4) Through any point one and only one line of a congruence can in
general be drawn.
To find the line through any point a of the congruence defined by the
group 0, notice that it must lie in the null planes of x with respect to any
two systems Si and S^ of the group. Hence xSi . xS^ is the common null
line through x.
Similarly in any plane X one and only one line of the congruence lies.
This Une is XS^ . XS^.
(5) The equation, xSi.xSt^^O, implies that a? is on one of the two
director lines of Q.
For if OiO^ and bjb^ are the director lines, and /Sfj = XiOxOs + fH&A,
/Sf, s X^a,+/ia&i&8} then xSi . x8^:== (\yfd^ ^ \fjLi) xa^a^. xbib^.
Hence, assuming that the director lines are not co-planar, either a:aiaB=0,
or xbib2 = 0.
Similarly the equation, XSi . XS2 = 0, implies that the plane X contains
one of the director linea
If xSi . xSt = 0, and XSi . XS2 = 0, then the theorems of subsection (4)
do not hold.
(6) If the congruence be defined as the assemblage of the director lines
of the quadruple group 0\ the line belonging to it which lies in any plane or
passes through any point can be determined thus :
Lemma. If L denote a single force the two equations, {abL)^0,
(&cZ)=0, imply the equation (caZ)=0 and that L lies in the plane abc.
But if L denote a system which is not a single force then the three
equations cannot coexist. For the equations (obL) = 0 and (bcL) = 0 imply
that b is the null point of the plane ahc with respect to L. Hence ca cannot
be a null line (assuming that aAc is not zero), unless L represent a single
force lying in the plane abc.
Now let ahc represent any given plane, and let XJS^' + \J3/ + X^Sg' + X^^'
represent any system of the group 0'. Then it follows firom the Lemma that
the three equations,
X, {bcS,') + X4 {bcS:) + X, (fccS/) + \ (bcS,') « 0,
\{caS/) + \{caS/) + \,(ca8,')'^\{caS^')^0,
X,(a6S,')+ X4(a6S/) + X.(a6S;) + X, (a6S/)« 0,
are the three conditions that this system may represent the director line in
the plane aic.
172]
QUADRUPLE AND DUAL GROUPS.
289
Hence the system of the group 0' which can be written in the form
S^% 1 oil y Ok • Oc
(6cS.'), {hcS,% {hc8:\ {hcS:)
{ca8^\ {caa:\ {caS;i (caS,')
(abS^'l {ab8:i (obS;i {abS,')
is the director force of the group which lies in the plane abc.
(7) Similarly the line of the congruence, which passes through any
point ABO, where A, B, C are planes, is found by substituting -4, B, C for
a, 6, c respectively in the above expression.
(8) Again, if the plane ahc contain one of the two common null lines
of 0\ then every line lying in it and passing through its point of intersection
with the other common null line must be a director line.
Hence the above expression for the single director line lying in the plane
ahc must be nugatory.
Accordingly the conditions, that the plane ahc may contain one of the
two common null lines of 0\ are
(6c/8f,0, {J>cS:\ QkS,'), (bcS.') =0.
(caS,% (caS:\ (caS/), (cafif/)
(abSO, {ab8:i (ohS,% {ab80
Similarly the conditions, that the point ABG may lie on one of the
common null lines, is found by replacing the points a, 6, c by the planes
A, By Cia the above conditions.
(9) An exceptional type of dual group arises, when the director equation
has two equal roots. In this case, with the notation of subsection (1), if Si
and S^ be any two systems of the group,
(SA) (SA) = (-SA)'.
A group of this type will be called a parabolic group.
There is only one director force in the group. Let it be Z), and sub-
stitute D for Si in the above equation. Then, since (DD) = 0, the equation
reduces to (DSi) = 0, Hence the director line is a common null line of
all the other systems of the group ; in other words, the director force is
reciprocal to every other system of the group.
The null plane of a point on the director line is the same for each
system of the group, and contains the director line. For, if fif be any system
of the group and D the director force, any other system of the group can be
written XD + fjJS, Hence, if «? be any point on the line D,
x(7J) + fjL8):=^fixS = xS.
Since the director line is a common null line of the group, the plane xS
contains the director line.
Similarly the null point of a plane containing the director line is the
same for each system of the group, and lies on the director line.
w. 19
290 GROUPS OF SYSTEMa [CHAP. IL
The theorems of subsection (4) still hold. For, if n; be any point not on
the director line, the common null lines of the group through x must
intersect the director force D ; and therefore must pass through the common
null point of the plane xD. Hence there is only one such line through x,
and there is always one such line. Also, if 8i and S^ be any two systems of
the group, the common null line through x is xSi . x8^.
The theorem of subsection (5) still holds. For, if ah be the director force,
any system of the group can be written in the form ac'\-hd.
Now xab .x{ax)-{- bd) = (xahc) xa + (xahd) xb.
Hence, xab . x {ac + hd) = 0, implies {xabc) = 0 = {xahd). Therefore x must
lie on the line ah.
Now, if fl^i = oc + hdy any other system 8^ of the group can be written in
the form Xa6 + fiS^.
Hence xS^ . xS^ = xSi . x (Xab + fiSi) = XxSi . xab.
Now X is not zero, if ^9 be different from 81. Hence, x8i.x8^^0,
implies, xab . x8i = 0.
(10) If iV be any line not intersecting the director force D of a parabolic
group, then one and only one system of the group can be found for which N
is a null line.
For let 8 be any system of the group. Then XD + /tt/S is any other
system. If iV is a null line of this system
X(ND) + fi(NS) = 0.
Now by hypothesis ( JV7)) is not zero. Hence the system D (N8) — 8 (ND)
has N for a null line. And no other system has N for a null line.
If D = 61^8, and N=^e^4, then the conjugate with respect to
D{N^8)-8{ND)
of the line «ie, must intersect both D and N, Hence D (NS) — 8 (JVD) can
be written in the form
XeiCt + fiabf
where CiCs is any given line intersecting D and N, and a lies on D and b
on N,
173. Anharmonic Ratio of Systems. (1) The null points of any
given plane with respect to, the systems of a dual group are coUinear. For
let the two systems 81 and /S, define the group, and let 8 be any third
system of the group. Also let A be any plane.
Then /S = XjiS^ + Xafif,, also the null point of A with respect to iSf is
A8 = XiA8i + Xgilfif,. Hence A8, AS^, AS^ are collinear.
173] ANHARMONIC RATIO OF SYSTEMS. 291
(2) The anbarmonic ratio of the four null points of any plane with
respect to four systems of a dual group is the same for all planes and
depends only on the four systems. For let 8i, S^, Xi8i + \^i, fhSi + fi^^ be
the four systems The four null points of any plane A are ASj, AS^,
XiASi + 'K^ASif fhASi + fh^8^' The anbarmonic ratio of these four points,
taking the first two and the last two as conjugates, is Xj/i^A^. This ratio
is independent of A.
(3) Similarly the four null planes of any point a with respect to the
four systems have the same line of intersection, and their anbarmonic ratio is
also Xi/%/X^.
(4) Definitions. Let this ratio be called the anbarmonic ratio of the
four systems. If the anbarmonic ratio be — 1, the four systems are said to
be harmonic ; and one pair are harmonic conjugates to the other pair. Pairs
of systems, harmonically conjugate to the two systems 8i and /S,, are said to
form an involution, of which Si and S^ are the foci.
The anbarmonic ratio of the four systems XiSi + X«S„ V^i + V^i*
V'Si + ViSf,, \''% + K% is
{\K - \K) (Vxr - X, V0/( W - x,V'0 (Xx V - x, V)-
(5) There is one and only one system belonging to a dual group which
is reciprocal to a given system of the group. For if UiSi + aJSi be any
given system, and X^Si + XjS, a system of the dual group reciprocal to it,
then
\ {«! (SiSi) + a. (8 A)} + X, {a, (SA) + «. (8M - 0.
And this equation determines Xi : X, uniquely. Thus a dual group can be
divided into pairs of reciprocal systems. Each director force is its own
reciprocal system.
But if the group be parabolic [cf. § 172 (9)], the director force is the only
system of the group reciprocal to any of the other systema For, if S be any
system and D the director force, any other system can be written \D + fi8.
If this system be reciprocal to S, X{D8)'^fi (88)^0, But (D8)^0, and
(88) is not zero. Hence /tt = 0.
(6) A pair of reciprocal systems of a dual group are harmonic conjugates
to the two director forces of the group.
For let A aiid J), be the two director forces, and XiA + X^s and
/ji^Di-h fij)t be the two reciprocal systems.
Then (X^ + X^) (A A) = 0.
Hence (assuming that the director lines do not intersect),
Xi/X, = - fiilfi^.
The two reciprocal systems can therefore be written XiA + XfDa,
XiA'X^,, and are harmonic conjugates to A s.nd A-
19—2
292 GROUPS OF SYSTEMS. [CHAP. II.
(7) Hence systems 8i, S^, 8^, etc., belonging to one dual group form an
assemblage of systems in involution with their reciprocal systems Si\ 8^', 8/,
etc,, belonging to the same dual group. The foci of the involution are the
director forces.
The dual group will be called elliptic or hyperbolic according as these
foci are imaginary or real.
(8) Since a single system uniquely defines a linear complex, we can also
speak of the anharmonic ratio of four linear complexes which have the same
congruence in common. An assemblage of complexes with the same con-
gruence in common contains two and only two special complexes. These are
the foci of an involution in which each complex corresponds to its reciprocal
complex, that is, to the complex of the assemblage which is defined by a
system reciprocal to its own.
These theorems respecting linear complexes are merely other statements
of the theorems proved above.
174. Self-Supplementary Dual Groups. (1) Let the operation of
taking the supplement be assumed to refer to any given quadric.
The system \8 will be called the supplementary system of 8, where 8 is
any system. Also 8 and |iSf define a dual group. This dual group has the
property that the supplement of any system belonging to it also belongs to
the group.
For if S' = Xj8f + /A|/S, then |flf' = X|S + /aS. Let the group be called
* self-supplementary.'
(2) A self-supplementary group is obviously in general determined by
any one system belonging to it. For if 8 be known, 8 and \8 in general
determine the group.
But if 8' be of the form \8 ±\\8, then \8' = ± 8\ hence 8' and \8' do not
determine the group. A system 8\ such that \8'=±8\ is called a self-
supplementary system.
(3) If two generators of the same system of any quadric are conjugate
lines with respect to any system of forces, then the generators of that system
of generators taken in pairs are all conjugate lines with respect to that system
of forces. Let 8 be the system of forces, Di and D, the two generators which
are conjugate with respect to 8 ; and let 0 be any third generator of the
same system of generators. We require to prove that ('S— o^>j^ Oj is also
a generator of the quadric.
Now let the operation of taking supplements be performed in reference
to this quadria Then |A=± A> and (A=±A, where both the upper
signs or both the under signs are to be taken [cf. § 116 (3)]. Hence since 8
can be written X^D^ + >*/)„ we have \8=±8. Also \0 = ±0.
174] SELF-SUPPLEMENTARY DUAL QBOUPa 293
Therefore
(«-^i^«)=i«-ip«
_ f l(iSflf) \
(OS)
Accordingly the conjugate of (? is a generator [cf. § 116 (8)].
(4) Conversely it is obvious that if £f be self-supplementary, that is,
if |/S= ±iS, then the conjugate of any generator 0 belonging to one of the
two systems of the self-normal quadric is another generator of the same
system of generators as O.
i^-iU'h^i'-^y
It is obvious that if |£f = fif, the generator must be of the positive system ;
if |/8f =5 — jS, the generator must be of the negative system.
(5) In general the director lines of a self-supplementary group are
supplementary to each other.
For if the group be defined by S and \S, the director equation is
Let the roots of this equation be aj/S and ff/a, then the director forces A
and Aare A = aS + /8|flf, A = /8S+a|S. Hence |A = A,and |A = A.
(6) But if we choose two director forces so that each lies on the self-
normal quadric, that is, so that | A = ± A» and | A = ± A (making the same
choice of both ambiguities), then any system jS = \A + mA belonging to the
group is self-supplementary. Hence these exceptional groups cannot be
defined by two systems of the form 3 and \8, Therefore the above reasoning
£uls.
Also if (SS) = i (jS|/Sf), the roots of the director equation are equal ; and
the group is parabolic [cf. § 172 (9)]. If (88) = (8\S), the director force is
8—\8, and is self-supplementary, and belongs to the negative system of
generating lines: if (flf/S) = — (S|iS), the director force is jS+|/Sf, and belongs
to the positive system. This is the most general type of self-supplementary
parabolic group, in which each system is not self-supplementary.
(7) In general there is one and only one self-supplementary system of
each type (positive and negative) in each self-supplementary dual group.
For if the group be defined by 8 and \8, where 8 is any system, or by 8'
and \8\ where \8' is any other system of the group, then any two pairs of
self-supplementary systems of the two types belonging to the group are
8±\8, and 8' ±\8'.
But if flf' = XjS + /li |S, then |S' = X |Sf -h /iS ; and hence
8'±\8' = {X±fi){8±\8)^8±\8.
Thus all such pairs of systems are identical.
294
GROUPS OF SYSTEMS.
[chap. II.
(8) Any system S which is not self-supplementary has in general two
and only two conjugate lines which are supplementary. The system obviously
has one pair of such conjugate lines, namely, the director lines of the group 8
and \S, It has no more, for if possible let D and \J) be two such lines which
are not the director lines of the group 8, \8.
Then 8^\D + fi\D;
hence |S = X|J) + /a2).
Accordingly D and \D must be director lines of the group 8, \8, which by
hypothesis is not the case.
This proposition does not hold, if the group (S, |iS) be parabolic.
(9) This proposition may also be stated thus : Any system has in general
one and only one pair of conjugate lines which are polar reciprocal to each
other with reference to a given quadric.
Let ab and cd be this pair of conjugates for any system 8, Let ab and cd
meet the quadric in a, b and c, d. Then ad, ac and bd, be are generating
lines of the quadric. But these lines are also null lines of the system 8.
Hence in general [cf. § 176 (12) and (13)] any linear* complex has four
lines which are generators of any given quadric, two belonging to one system
of generators and two belonging to the Other system.
(10) The proposition can easily be extended to self-supplementary
systems with respect to the given quadric. For if 8 be any system, then
/Sf ± |/Sf is the general type of a self-supplementary system. But the director
lines of the group 8 and \8 are supplementary, and they are conjugate lines
of 8 ±\8 which belong to the dual group.
The discussion of self-supplementary systems, and of systems such that
(88) ± (S |/Sf) = 0, is resumed in § 175 (8) to (13).
* Cf. Clebsoh and Lindemaun, Vorle$ungen Uber Oeometriet vol. n.
175] TRIPLE GROUPS. 295
176. Triple Groups. (1) The reciprocal group of a triple groap is
another triple group. Let Si, 8t, S^ define any triple group (?, and let
Si, Si, 8^' define the reciprocal group 0\ The director equation of 0,
namely, the condition that \iSi + XA + X,Sf, reduce to a single force, is
This equation is also the condition that the line Xi8i + XjSt -f X^fif, be a
common null line of the group 0\
(2) The condition that x may lie on a common null line of 0 is,
(x8i.x8^.8^)r=0.
For x8i . x8^ is a common null line of 8i and 8^ and the given condition
secures that it be also a null line of 8^.
(3) But the equation, (x8i.xSt.8z) — 0, is the equation of a quadric
surface.
Hence the common null lines of a triple group 0 are generators of a
quadric sur&ca The director lines therefore, which are null lines of the
triple group Q', must also be generators of a quadric surface. Furthermore
every nuU line intersects every director line, and conversely. Thus it follows
that the quadric surfeu^es, on which the null lines of 0 and of O' lie, must be
the same surface ; and that the null lines of 0 are generators of one system
on the surface, and the director lines of O (i.e. the null lines of 0^ are
generators of the other system on the surface. Let the two systems of
generators be called respectively the null system and the director system
with respect to the given group.
(4) Hence a triple group 0 defines a quadric surface. The only other
triple group which defines the same surface is the reciprocal group 0\ The
director system of generators with respect to & is the null system with
respect to 0\ and vice verscu
(5) Conversely, any quadric sur&ce defines a pair of reciprocal triple
groups.
For take any three generators of the same system belonging to this
quadria Let Oi, G„ &» be forces along them. Then Oi, &,, (?, define a
triple group, and its associated quadric must contain the three lines
Ou Oi, Oz- But there is only one quadric which contains three given lines.
Hence the associated quadric is the given quadric.
(6) The condition that the plane abc may contain a director line of the
group 0 is
{bcSil (6cS,), (bcS,)
{caSil (coflf,), (caS,)
(abSi\ {ab8,\ (abS,)
= 0.
296 GROUPS OF SYSTEMS. [CHAP. II.
For assume that XiS^ + X^, + X,fif, is a single force lying in the plane ahc.
Then, by the lemma of § 172 (6), the three following equations are the
necessary and sufficient conditions,
Xi (6cSi) + Xs (6cS.) + >^ (ftcflf,) = 0,
\ (caSi) + X, (caS;) + X, (caSs) = 0,
Xi (oJflfi) + X, (abS;) + X, (abS^) = 0.
But these equations require the given condition.
Accordingly this is also the condition that ahc may touch the associated
quadric and contain a common null line of the group.
(7) Similarly the condition that the point ABG, where A, B, C are
planar elements, may lie on the associated quadric is found by replacing
a, b, c in the above condition by A, B and C,
(8) If the supplements of 0 and 0' be taken with respect to the asso-
ciated quadric, then from § 174 (3) and (4) every system belonging to G or
0' is self-supplementary; and conversely all self-supplementary systems
with respect to a given quadric must belong to one of the two associated
groups of the quadric.
For any system 8 of group 0 we may assume |S = S; then for any
system S' of G' we have \S' = - 8'.
(9) Corresponding to each director line of a triple group, one parabolic
dual subgroup can be found with that line as director line.
For let -^1,-^8, F^ be any three director lines of the triple group, and let
Fi be the given director line. Then any system 8 of the triple group can be
written
8 = XiFi + XjF, + XjFj.
Now, if the subgroup (^i, 8) is parabolic, {Fj,8) = 0. Hence the required
condition is
X,(FiF,) + X,(F,^,) = 0.
Thus the subgroup defined by F, and (F^F^) F^ - (F^F^) F^ is parabolic
with ^1 as director line.
(10) Let the quadric defined by the triple group be self-supplementary.
Hence by the previous subsection, if 8i=(FiF;,)F^''{FiF^)F3, the dual
group defined by Fi and 8^ is parabolic and such that each system 8 is self-
supplementary. If|i?\ = i^i, then|/8f=S; andif |i^i = -Fi, |S=-flf. Corre-
sponding to each generator of either system there is one such parabolic
self-supplementary dual group [cf. subsection (12), below].
(11) The most general type of self-supplementary parabolic group, in
which each sj'stem is not self-supplementary, is the type defined by a
generator, 0, of the self-normal quadric and a self-supplementary system 8 ;
such that, either |G = (7, and |S = -£f, or, |(? = -G, and \8^8[ct. § 174 (7)].
175] TRIPLE QROUP& 297
For firstly let 0, the director line of the parabolic subgroup, be such that
I ff s= G ; and let 8i be any other system of the group. Then by § 174 (6)
Also by hypothesis, (G8^ = 0.
Then any system S of the group can be written \0 + /*/Si.
Hence |5 = X |G + /ii |flf=(X + /ia)G-/i/Sfi.
Thus, if jS be self-supplementary, that is, if J 5 = 5, then, \ + /Aa = — X;
that is, X = ^ ^/Ao.
Hence the system flf = Sj — JaG, is such that |flf = — S.
Accordingly the self-supplementaiy parabolic group can be defined by
GandS; where |G = ff. |S = -fif.
Similarly if | (? = — (?, then the self-supplementary system 8 belonging to
the group is such that \S^S,
Thus corresponding to any generator 0 of the self-normal quadric there
are an infinite number of such parabolic self-supplementary groups, since any
self-supplementary system 8 of the opposite denomination (positive or
negative) to 0 will with 0 define such a group.
(.12) It is evident [cf. § 174 (3) and (4)] that any self-supplementary
system 8 has as null lines all the generators of the self-normal quadric of the
opposite denomination. It also has as null lines two generators of the same
denomination.
For we may write 8 = aj)i + o^Z),, where A a^d J), are two generators of
the same denomination as 8, Let D, be a third such generator. Then any
self-normal system of the same denomination as 8 can be written in the
form
This system is a generator (D) if
X,X, (A A) + ^\ (DJ)i) + \^ (A A) = 0.
Also D is a null line of S, if (DS) = 0, that is, if
(\a, + \a) (A A) + X, {«! ( AA) ^^ (DJ>b)] = 0.
These two equations give two solutions for the set of ratios of X^ to X, to
Xs. Hence 8 has two null lines among the generators of the same denomi-
nation [cf. subsection (10) above, and also § 174 (10)].
(13) Now let N he a, generator of the opposite denomination to the
self-supplementary sjrstem 8 ; and let D and D' be the two generators of the
same denomination as 8, which are null lines of 8 according to the previous
subsection.
Then D and 1/ necessarily intersect N. Also the parabolic group
defined by N and 8 is of the type discussed in subsection (11). But D and
D' and N must be common nulls of this group. Also no other generators of
298 GROUPS OF SYSTEMS. [CHAP. 11.
the quadric can be null lines of any system of the group, other than N and
S. For consider the system \If+fi8. Then every generator of the D type
intersects iV, but only D and D' are null lines of S. Accordingly only D
and ly of the generators of this type are null lines of XN+fi8. Again,
all the generators of the N type are null lines o( S; but no generator of this
type, except JV, intersects N. Hence N is the pnly null line belonging to
the generators of this type.
Hence any system S\ not self-supplementary, which is such that
{8'S') ±(8' \8') = 0, has two generators of one system and one generator of
the other system as null lines. This proposition should be compared with
that of § 174 (9).
(14) Thus, summing up and repeating, any quadric has in general two
generators only of one system and two generators only of the other system,
which are null lines of any system of forces 8. But, as exceptional cases,
either all the generators of one system and two only of the other system are
null lines of 8 ; or one generator only of one system and two only of the
other system are null lines of 8.
176. CJONJUGATE SETS OF SYSTEMS IN A TRIPLE QrOUP. (1) Any two
systems 8i, 8t of the triple group O define a subgroup. It is possible to find
one and only one system 8 belonging to 0 which is reciprocal to the whole
subgroup 8i, 8t,
For let 8 be such a system and let £f, be any third independent system so
that 8i, 8if 8^ define G. Then we may vrrite 8 = X^Si + XA + X,S,.
Hence by hypothesis
\ {8 A) + X, (8,8,) + X, (8 A) = 0,
\ (8J3,)+\,(8A)+MSA)=^0.
Thus the ratios Xi : X, : X« are completely determined, and therefore 8 is
completely determined. The reciprocal system 8 can be written in the form
^, 8ty 8z
%8,\ {8A)> (aA)
(8 A), (8AI {8A)
This system does not belong to the dual subgroup (8i, 82), it the
coefficient of 8^ does not vanish ; that is, if
be not zero; that is, if the subgroup (81, 8^) be not parabolic. In subsections
(2), (3), (4), following, the subgroups will be assumed to be not parabolic.
(2) Also in the subgroup defined by 81, 8^ we may choose 81 and 3^ so
as to be reciprocal [c£ § 173 (5)]. Thus three systems £fi, 8^, 8^ can be
found, belonging to the triple group O, such that each system is reciprocal
176] CONJUGATE SETS OF SYSTEMS IN A TRIPLE GROUP. 299
to the subgroup formed by the other two. And one of these systems, say Si,
can be chosen arbitrarily out of the systems of the group 0 ; and then S^
and 8t can be chosen in a singly-infinite number of ways out of the dual
subgroup of 0 which is reciprocal to 8i,
Definition. Let such a set of three mutually reciprocal systems of a
group 0 be called a ' conjugate ' set of the group.
(3) If 8iy 8^, Sthe a conjugate set of systems, then XSi, X8t, XS^ are
three conjugate points lying in the plane X with respect to the associated
quadric of 0.
For the director lines of the group (Si, 8^) are generators of the quadric
G; and it has been proved [cf. § 173 (6)] that the line joining the points XSi
and XSi intersects these director lines in two points di and (2, such that the
range formed by (c2i, (2,, XSi, XS^) is harmonic But di and d!, are on the
quadric 0. Hence by the harmonic properties of poles and polars XSi is on
the polar of XSf, and XS^ on the polar of XSi.
Similarly for XS^ and Xfif,, and for XSi and XS^. Hence the three
points XSi, XSu XSt are three mutually conjugate points on the plane X.
(4) An analogous proof shows that tcSi, a?£f,, xSt are conjugate planes
through the point x.
CHAPTER III.
Invariants of Groups.
177. Definition of an Invariant. (1) Let 8u 8^, ... Sp define a
group 0 otp — 1 dimensions, and let Si, S^', . . . 8/ be any p systems belonging
to this group 0. Then there must exist p equations of the typical form
Also let A denote the determinant
^pl> ^pB» ••• ^P(»
Then, if A be not zero, the systems 8i, <8^', ... flfp' are independent [c£ § 96
and § 63 (4)] systems.
(2) Let <f>{8i, Si, ... 8ft) be any function of the p systems 8^ 8^, ... 8p
formed by multiplications and additions of Su 8^, ... Sf, and of given points,
forces, and planar elements. Let ^(fi^/, 8^, ... 8^) denote the same (unction
only with S/, 8^', ... 5/ substituted respectively for Si, 82, ... 8p.
Then if (l>{Si\ 5,', ... 5^0 = ^''* ('Si, S„ ... Sp), \ being an integer,
<l>(8i, i8^, ... jSp) is called an invariant of the group 0.
The effect of substituting any other p independent systems of the group
0 for 81, £>,,... Sf, in an invariant of the group is to reproduce the original
function multiplied by a numerical fiEU3tor which does not vanish.
178. The Null Invariants of a Dual Group. (1) Let 81 and S,
define a dual group, and let 8^X81 + fiS^, 8' = X'Si + /a'S„ A = V' - XV
Then the expressions xSi . X82 and XSi . XS^, where x is any point and
X is any planar element, are invariants of the group. Call them the Null
Invariants.
For wS . wS' ^ AxSi . xSi, and ZS . ZS' = AZS^ . ZS^.
It has already been proved [cf. § 172 (4)] that these expressions denote
respectively the common null line of the group through the point x, and
the common null line of the group in the plane Z.
CHAP. III. 177 — 179] THE HABMONIG INVABIANTS OP A DUAL GROUP. 801
179. The Harmonic Invariants op a Dual Group. (1) Another
important invariant of the group is aSi . 8^ — ocSt . Si. Call this expression
the Harmonic Point Invariant of the group ; let it be denoted by H{x).
This expression is easily proved to be an invariant by direct substitution.
It represents a point. It must be noticed that the intensity to be ascribed
to H(x) depends on the special pair of systems (£>,, 8^) which is chosen to
define the group.
It is obvious that H (Kx + fjLx') = \H (a?) + /JST (x).
(2) Similarly if X be any planar element, X8i . 8^ — XS^ . jS^ is an
invariant of the group. Call this expression the Harmonic Plane Invariant ;
and let it be denoted by H{X). It represents a planar element. The
intensity of H{X) depends on the special pair of sjrstems which define it.
Also n{\x + /iZO = xfl^(Z) + iiH{xy
(3) If 8i and £1^ be a pair of reciprocal systems, it is obvious from
§ 167 (5), equation 25, that
x8i . 82 + x8^ . Sj = 0.
Hence in this case H(x) = 2a?iS, . Sj = — ix8^ . S,.
Similarly H{X) = 2Zft .8^— 2X8^ . S,.
These expressions only hold when 81 and 8% are reciprocal.
(4) To find the relation between the points x and H(x), and between
the planes X and H{X).
Let the common null line through x meet the director lines of the group
in di and d^ ; and let the two director lines be written diBi and d^.
Then 8^ and 8^, which will be assumed to be reciprocal sjrstems, can be
written in the forms [cf. § 173 (6)]
Also we may write x = ^idi + f 2^.
Hence by multiplication x8i = l^idid^ + i^l^e^
x8i.82= X^idid^ . diBi - \^4Aei .d^ = \ {d^e^d^ {fidj - fad,}.
Also {8 A) = 2 (di^dA), (8 A) = - 2X» {d,e4^.
Therefore il(a?)=ac5,.5, = V{-(fl^A)(S;S,)}(fA-fA).
But fidi+fsd^, fA — fA> di, da form a harmonic range. Hence H(x)
lies on the common null line of the group through x^ and is the harmonic
conjugate of x with respect to the two points in which the null line meets
the director lines.
(5) Similarly ir(Z)=2Xflfi.iS«= V{-(SiflfO(/SA)}(fiA-fA); where
Di and D, are two planes both containing the common null line in the plane
Xf and respectively containing the two director lines ; and X = f ,i)i + fa^a*
302 INVARIANTS OF OBOUFS. [OHAP. III.
Hence H{X) contains the common null line of the group which lies in
X, and is the harmonic conjugate of X with reference to the two planes
containing the null line and the two director line&
(6) Let H[H{x)] be written H*{x\ and let H^{x) denote H[H^{x)],
and so on.
Then it has been proved in (4) and (5) that if di and d^ lie on the director
lines of the group, and x = fjdi + f^d,.
H {x) = V{- {8 A) (S^.)] (f A - f A).
It follows that H^ (x) = - (8 A) (SA) (^A + f A) = - (8,8i) (SA) ^.
Therefore H*{x) = x, and generally H^(x) = x, or =H(x), according as \
is an even or an odd integer.
Similarly JJ» (X) = - (8 A) (SAi) X ; and hence IP (X) = X.
(7) If the group be parabolic [cf § 172 (9)], then H{x) is the null point
(common to all the systems) of the plane through x and the single director
line. For let D be the director force and 8 any other system of the group,
then (DS) = 0.
Hence by subsection (3), H (x) = 2xD . 8.
Thus H (x) is the null point of the plane xD with respect to 8.
Accordingly all the points of the type H(x) are concentrated on the
director line ; and, if (xyD) = 0, then H{x) = H (y).
Similarly H(X) is the null plane of the point DX.
180. Further Properties of Harmonic Invariants. (1) If 8i and
8^ are two reciprocal systems of the group, the null plane of x with respect
to S| is the same as the null plane of H(x) with respect to 8^. For by
§ 167 (2), since x8i and x8t are planar elements,
H(x)8,=^ 2x8,. 8,. 8,= (8A)ai8,\ ,^,
and H(x)8, = ^2x8,.8,.8, = -(8A)a^S^^ ^ ^'
Similarly the null point of X with respect to 8, is the same as the null
point of H{X) with respect to 8^.
For H{X)8,^ 2X8,. 8,. 8,^ (8A.)X8,\ ,.,
H{X)8,^-2X8,.8,.8,^'-(8,8,)X8^.] ^^^•
(2) If flf be any system of the dual group, to prove that
H(x)8=-'H(x8), H(X)8^'^H{X8) (3).
For let 8' be the system reciprocal to 8 belonging to the group. Then
we may write H(x)=s2x8 .8'. Also from the second of equations (1) in
subsection (1),
H(x)8^-{88)x8\
Again by § 167 (2) H(x8) = 2x8 .8.8'^(88)x8'^^ H(x)8
Similarly H {X8) ^'H(X) 8,
180, 181] FURTHER PROPERTIES OF HARMONIC INVARIANTS. 303
(3) If the locus of a; be the plane X, then the locus of H{x) is the plane
H{X).
This proposition is obvious from the harmonic relation between x and
H{x) and between X and H{X).
It can also be proved by means of the important transformation
XH(x)^xH(X) (4),
where x and X denote respectively any point and any plane.
For if 8i and 8^ be any two reciprocal systems of the group, then
remembering that the product of two planai* elements and a force, or a
system of forces, is a pure regressive product,
XH(x)^2X.(x8,.82)^2X.xS,.S^^-2xSi.XS^
= - ac(flfi . Z/S,) « 2x(X8, . 8^) = xH(X).
(4) If a& be a null line of any S}rstem 8 of the dual group, then
H(a) H(b) is also a null line of 8.
For by hypothesis {ah8) = 0. And by (2) of this article,
H{a) H(b)8== -H(a) H{b8).
But by (3) of this article and by § 179 (6),
H (a) H(bS) = b8H^(a) = b8a = 0.
Hence H(a)H(b)8 = 0,
Since H*(x) = x, this proposition can also be stated thus, if aH(b) be a null
line of 8, then bH (a) is a null line of 8.
(5) If 8i and 8^ be reciprocal systems of the dual group and oi be a
null line of 8^, then -fir(a) H{b) is the conjugate of ah with respect to 8i.
This proposition will be proved [cf. § 164 (4)] by proving the important
formula
H{a)H(b)^-2(8A)aJS,.b8, ....(5);
where H (x) = 2x8^ . 5«.
For remembering that {818^) — 0, and (o&jS^) = 0, and twice using equations
(22) of § 167 (3),
H(a)H(b)^*(a8,)8^.(b8,)8^^*(a8i.b8,.8^)8,-2(8^,)a8,.b8,
--*[{((^S,)8,^^{8,8,)ab}8,]8,-2(8A)aS^.b8,
= ^2(8^;)a8^.b8,.
In connection with this proposition and that of subsection (4) the proposition
of § 166 (2) should be referred to.
181. Formula connected with Reciprocal Systems. (1) A variety
of formulse connected with two reciprocal systems can be deduced from the
preceding article.
Thus equation (22) of § 167 (3) can be written
(ab8)8=:^^(88)ab-^a8,b8.
304
INVARIANTS OP GROUPS.
[chap. ni.
From this equation and from equation (5) of § 180 (5), it immediately follows
that, if 8i and 8^ be reciprocal and a6 be a null line of 3^,
Similarly, 2 {ABS^) 8, = (8 A) AB - ^^ H{A)H (B) ;
where AB is a null line of jSs*
(1).
(2) Also with the same assumptions as in (1), it follows fix>m § 180(5)
that aH(b) is a null line of 8i. Hence by the preceding subsection
2 [aH (6) 8,} 8, = (8A) aH(h) - ^g^^S(<^) H* («)•
But by § 179 (6), H*(b) (8 A) (8^t) b ;
also by an easy tiansfonnation
{aH(h)8,} = i8A)(ab8,).
Hence 2{ab8i)8,='aH ib)-bH(a). 1
Similarly 2(AB8i)8t = AH{B)- BH(A).)
(3) Also, since {ab8t) = 0, (abS,) 8, = 0 = ^ (8 A) ah + a8,. 6/8;.
2
(2).
Hence
a6 = —
(SA)
OOs • OOj.
(8A)
Thus (018,) 8, = i (iSi/S,) a6 + a8i . b8, = a8i . b8, - ^^. cuSf,. 6/S,.
(SiiSO
...(3).
Similarly, if {AB8;) = 0, (AB8,) 8, = Aft . B8, - )^ ilft . B8^.
(4) Also, with the same assumptions, equations (26) of § 167 (5) become
(a6ft)ig, = cuSfi.6ig, + aS,.6ft, ) .
(AB8,)8,^A8,.B8, + A8^.B8,.] ^*^'
182. Systems reciprocal to a Dual Group. (1) Let R be any
system reciprocal to a whole dual group. Then R belongs to the reciprocal
quadruple group. Also let 8, and ft be two reciprocal systems of the dual
group.
Then by equation (26) of § 167 (5) and remembering that (iJft) = 0 = (R8^X
fl^(ZiJ) = 2(jrB)ft.ft = -2(Zft)i2.ft = 2(Xft)ft.i2 = ir(Z)iJ.
Similarly, H(xR) = H{x) R.
(2) We may notice by comparison of this result with § 180 (2) that if
8 be any system of the dual group,
H(w8)^-H(x)8, H{X8)^-E(X)8.
But if i2 be any system of the group reciprocal to the dual group,
H(xR) = H(x)R, H(XR)^H{X)R.
182, 183]
SYSTEMS RECIPKOCAL TO A DUAL GROUP.
305
183. The Pole and Polar Invarunts of a Triple Group. (1) Let
the triple group 0 be defined by three sjrstems Si, S^, S,. The same three
systems taken in pairs define three dual subgroups. Let these dual subgroups
be denoted by gi, g^^ g^ ; thus, let the group g^ be defined by S^^ S,, the group
g% hy flf,, Si, and the group '^r, by Si, 5a .
Let the harmonic invariants of the point x or of the plane X with
respect to the groups giy g^ and g^ be denoted respectively by Hi(x)j Hi(X\
H,{x). H,iX). H^ix), H,{X).
(2) The expression
. ff, («) .
8i , 82 , S3
(8,8,), (<SA), (-S^,)
(8A). (8^,). i8A)
(1).
(8 A), {8 A)
(8A). (SA)
will be proved to be an invariant of the group, and will be called the Polar
Invariant with respect to the group 0. Similarly the expression
(8A), {8A)
. H, (X) .
Si , Sa , Ss
(SA), (SM (s^t)
(SM (>SA), (SA)
(2),
will be proved to be an invariant of the group, and will be called the Pole
Invariant with respect to the group 0.
(3) If -Bi be the system of the group 0 reciprocajl to the subgroup ^i,
then by properly choosing the intensity of Ri we may write [cf. § 176 (1)]
(SA), (SA)
(SA), (SA)
i?i=
Si , Si , s,
(fi^A), (SA), (SA)
(SA\ (SA), (SA)
Hence the polar invariant of x with respect to G is Hi(x)Iti, and the pole
invariant of X with respect to G is -Hi (X) iJ,.
Let the polar invariant be denoted by P(x) and the pole invariant by
P(Z).
Then P(x) = H,(x)Ri, and P (X) = J7, (X) i2j.
(4) Another form for P (x) and P (X) can be found as follows.
We have fl, (x) S^^lxS^.S,- «/S, . 8,} 8^ = {(8 A) x-2xSt.82]8,
= {8A)a'8,-(8J3,)a;8t.
Also H, {x)83={2x8,.S»- (8 A) a?} 8t = (8 A) «^ - i8A) «-»..
Hence from equation (2), P (x) = fl, (x) 8, - (8 A) xS, + (8,8^ x8,.\ . .
Similarly P (Z) = H, (X) S, - {8A) X8, + (8,8,) Xfl^.J * ' '^^^
(5) The invariant property can easily be proved from this latter form.
For write x {8,, 8,, 8,} for P(x) as defined above, in order to bring out the
w. 20'
S06 INVARIANTS OF GROUPS. [OHAP. III.
relations of P (x) to the three systems 8i, Sj, S,. Then it follows from the
form for P (x) given in equations (8) that
a? {8„ Sa, 8,}^-x{S,, S„ 8,} (a),
also x{8i,S^,8^}-=0 (6).
Furthermore H^ (a) 8, = {2x8^ . 8^ - (8^^) a?} 8i
= 2 (/gA)a?Sa- arSa. 5, . 8s-(8^s)^8^.
Hence x {8,, 8^, 8,}=^{SA)a>8^-^8^'8,, 8,-(8^,)x8^ + (8,S^)x8^
= (5A) ^fifa- (^A) ^fi^i - {2^/Sa . 8, - (flfjiSg) a:} S,
= H,(x)8,'-(8^,)x8, + (8A)(c8,
= ^{S„Si,fl^} (c).
Lastly x{8, + 8,\ 8,, 8,} = x{8u S„ Ssl + ^fS/, 8,, 8,} (d).
Now let 8, S', 5" be three systems of the group, such that
8=^'\£, + fi8, + v8,, S'^\% + M''8i + v%, 8^'^\'% + fi'% + i/'8s;
and let A denote the determinant S ± \fiv". Then from the equations (a),
(b), (c), (d), which have just been proved, we deduce at once that
x{8,8^,8"} = Ax{8,,8^,8,}.
This proves that P (x) is an invariant of the group.
An exactly similar proof shews that P (X) is an invariant of the group.
Now that the invariant property is proved we may abandon the notation
X [8,, 8^, 8,} tor P (x).
184. Conjugate sets of Systems and the Pole and Polar
Invariants. (1) Let iii, R^, iZg be a set of conjugate systems of the
group Q. Then
Also let gi, g^, g^ denote the subgroups R^Rt and R9R1, and RiR^
respectively. Hence Ri is reciprocal to the group gi, and R^ to the group ^2>
and Ri to the group g^.
Then P (x) and P (X) take the simple forms 2xR2 .R^.Ri and
2Xi2a . iZ, . i2i. This follows at once from the forms for P(x) and P(X)
given in § 183 (4), equation (3).
(2) It also follows that
P (x) = 2xR^ . ii, . iJi = 2xR^ . 12a . i?8 = ari?8 . IZx . IZa = - ZxR^ .R^.R, = etc. ;
with similar transformations for P (X).
(3) The equation, ojRi = P (6), can be solved for a. For let
Then multipljdng each side of the given equation by -Bi,
aRi.R,^^{R,R,)a^P(b)R,^2bR,.Rt,.R,.R,^(R,R,)bR^,Rt,.
Hence a = 2bR^ . i2j = - 26i?, . R^,
184, 186] CONJUGATE SETS OP SYSTEMS AND POLE AND POLAR INVARIANTS. 807
Also a condition holds. For
Thus (obR^) = - (baR^) = (RJt^) (bbR^) = 0.
Similarly (abR^) = 0.
Accordingly ab is a common null line of the subgroup gi.
186. Interpretation of P(x) and P(X). (1) P(x) denotes a
planar element, and P(X) denotes a point.
To find the plane P(x), write P(x) in- the form 2ari2a.i2,.i2i, which is
given in the last article.
Let the common null line through d? of the subgroup gi intersect the
director lines of gi in d^ and di'. Then di and rf/ are on the quadric 0
[cf. § 175 (4)]. Also the four points x, ixR^.R^, d^ d^ form a harmonic
range [cf. § 179 (4)]. Hence the point ixR^ . R^ lies on the polar plane oix
with respect to this quadric.
But P{x) is the null plane of this point with respect to Ri\ and therefore
the plane P (x) passes through the point 2xRi . i2,.
Now let Ri\ R^, R^ be another set of conjugate systems of the group 0.
Then the same proof shews that the plane P{x) passes through the point
2xR^ . R^'\ and that this point, 2xR^ . R^^ lies in the polar plane of x with
respect to the quadric 0, Similarly for a third set of conjugate systems,
such as iZx", Ri', R,".
Hence the plane P (x) passes through the three (not coUinear) points
arJRa.JJ,, 2xR,'.R^\ 2xR,'\R^'\
Hence P (x) denotes a planar element of the polar plane of x with respect
to the quadric 0. Similarly P{X) denotes the pole of the plane X with
respect to the quadric 0.
(2) It follows as a corollary from subsection (1) and from § 176 (3) and
(4) that we can express the angular points of tetrahedrons self-conjugate
with respect to the quadric 0, which have one face in a given plane.
For let X be the given plane, and Ri, R^, R^ a set of conjugate systems
of the group 0. Then by § 176 (3) XRi, XR^, XR^ are three conjugate
points in the plane X, and by the present article P(X) is the pole of X.
Hence these four points are the corners of a self-conjugate tetrahedron with
one face in the plane X,
By taking different sets of conjugate systems an infinite number of such
tetrahedrons may be found.
(3) Similarly we can express the four planes which are the &ce8 of a
self-conjugate tetrahedron with respect to 0, of which one comer is at a
given point x.
20—2
308 INVARIANTS OP GROUPS. [CHAP. III.
For, by the same reasoDing as that just employed, the four planes are sdRu
ar-Ra, xRi and P(a?).
By taking different sets of conjugate systems an infinite number of such
tetrahedrons may be found.
(4) The interpretations of P (x) and of P (X), which are given in (2)
and (3), shew that P^(x) [i.e. P {P(a?)}] must denote the point on, and that
P* (X) must denote the plane X.
This result can also easily be proved by direct transformation.
(5) Again it follows from the interpretations of P (x) and P (X) that if
y lie on P(x), then x lies on P(y); and that if Y contain P(X), then X
contains P(Y).
This result can also be proved by direct transformation, namely the
following equations hold
[P(x)y]=^[P(y)xl [P(Z)r| = [P(F)Z].
186. Relations between Conjugate Sets of Systems. (1) It fol-
lows from § 181 (3), equation (3), that if Ri, R^, Rshe a conjugate set of
systems, and if (abR^) = 0, then
(abR,)R, = aR,.bR,-^^^aR,.bR,.
Now if we take aRi^P(b\ then by §184(3) the condition (a6ii,) = 0 is
fulfilled ; and (aR^) = - (RA) bR^.
Also (obR,) = - (baR,) = - {bP {b)\ = {P (b) b).
Hence finally if b be any point,
{P (6) b}R, = P (b) , bR, - {R,R,) 6iJ, . bR^ -;
and, since b bears no special relation to jRi, by the cycKcal
interchange of suffixes, y (1)
{P(b)b}R^^P(b).bR,--(R,R^)bR,.bR,,
{P(b)b}R,=^P(b).bR,^(R,R,)bR,.bR,.j
(2) Similarly if B be any plane,
{P(B)B]R,^P{B),BR,^(R,R,)BR^.BR,;
{P{B)B}R^ = P(B),BR,^(R,R^)BR,.BR,\ (2)
{P(B)B]R,^P(B).BR,^(R,R,)BR,.BR,.^
(3) It is to be noticed that P(B), BR^, BR^, BR^ are the four angular
points of a self-conjugate tetrahedron with respect to the quadric 0. This
tetrahedron has the plane of one face, namely B, arbitrarily chosen, but is
otherwise definitely assigned by the conjugate set of systems Ri, R^, i2,.
Similarly, P(6), bRi, bR^, bRf are the four faces of a self-conjugate tetra-
hedron with respect to the quadric 0, This tetrahedron has one angular
point, namely 6, arbitrarily chosen, but is otherwise definitely assigned by
the set Ri, R^, R^.
186] RELATIONS BETWEEN CONJUGATE SETS OF SYSTEMS. 309
(4) Itet pi, P29 pt, p denote the angular points of a self-conjugate tetra-
hedron with respect to the quadric 0. Then one reciprocal set of systems
with respect to the group 0 can be expressed by
PPi + fhPiPzy '
PPii + f^PsPi, ' (3)
PPi + FhPiPi',.
where fh, fi^, fh are given definite numbers.
Similarly if Pi, Pj, P„ P denote planar elements in the faces of a self-
conjugate tetrahedron, then one reciprocal set of systems can be expressed by
PP« + X,P,Pi,i (4)
PPz + \PiP.l]
where Xi, A-j, Xj are given definite numbers.
(5) The proposition of the preceding subsection, symbolized in equations
(3) and (4), may be enunciated as follows : Corresponding to any given set of
conjugate systems of a group 0, one and only one tetrahedron self-conjugate
with respect to the quadric 0 can be found with three comers in a given
plane, such that its opposite edges taken in pairs are respectively conjugate
lines of the three systems of the conjugate set.
Also corresponding to any given set of conjugate systems of a group 0,
one and only one tetrahedron self-conjugate with respect to the quadric 0
can be found with one comer given, such that its opposite edges taken in
pairs are respectively conjugate lines of the three systems of the conjugate
set.
Such self-conjugate tetrahedrons will be said to be associated with the
corresponding conjugate sets of systems, and vice versa.
(6) The group 0[ reciprocal to the group 0 is also a triple group, and
defines the same quadric as 0.
Now let p, Ply p.2y pt be the four angular points of a tetrahedron which is
self-conjugate with respect to this quadric. Also let the conjugate set of
systems of the group 0 associated with this tetrahedron be
PPl + fhP^Pty PPi + fhPsPl* PPs + fhPlP2'
Then it is obvious that the conjugate set of the reciprocal group 0', associated
with this tetrahedron, is
PPi-fhPtP$, pp2-fhPtPi, PPs-fhPiP2'
For it follows from mere multiplication that any system of the last set
is reciprocal to each system of the first set. Hence the three systems of
the last set each belong to the group Q\ Furthermore they obviously are
reciprocal to each other, and therefore form a conjugate system of the group
0\ And lastly, the form in which they are expressed shews them to be the
conjugate set of systems associated with the tetrahedron p, Pi,p2,Pz'
310 INVARIANTS OF GROUPS. [CHAP. III.
(7) Similarly an analogous proof shews that if P, Pi, Pg, Ps be the four
buces of a tetrahedron self-conjugate with respect to the quadric of G and G\
and if the associated conjugate set of (? be
PP, + ,i,P,P,, PP, + fi,P,P,, PP, + fi,P,P,;
then the associated conjugate set of the group 0' is
PP,^fi,P,P,, PP,-fi,P,P,, PP,^ij^P,P,,
187. The Conjugate Invariant of a Triple Group. (1) If S^ S^,
flf, be any three systems of the group (?, the equation of the quadric 0 is
[cf. § 176 (3)] (xSi , xS^ . S,) = 0.
(2) If a and y be any two points and \a? + fty be a point on the quadric
lying on the line joining them, then
V (xSi . xS<, . S,) + \fjL {{xSi . yfifa . flf,) + ( yS, . x8., . 8^)} + fi^ ( yS^ . yS^ . 8^) = 0.
Hence the condition that the points x and y should be conjugate is
(x8i . y8^ . /S,) + (y8j . x8^ . S,) = 0.
If y be regarded as fixed, this is the equation of the polar plane of y.
(3) Let the expression ^ [x8i . y8.2 . 8^ + ySi . X82 . S^} be denoted by
0 (xy). It will be proved to be an invariant of the group 0, and will be
called the Conjugate Invariant.
The equation 0 (xx) = 0, is the equation of the quadric 0»
It follows from symmetry that, Q{xy) = Q(yx).
(4) In order to prove the invariant property let us write 0 (xy) in the
form ay {Sx, 8^, S,}.
Then obviously
«y {Si + «i', flf„ 8,}^xy[8u fif,, 8^}i-xy{S,\ S,. S,} (1).
Also a^ {8i, 8,, 8,} = - xy [8,, 8,, 8,} (2),
and xy{8y, S„ S,} =0 (3).
Furthermore y8i . S, = (838^) y — yfi^, . iSj.
Hence x8i.y8^. 8^^x81. (y8^.8;) = (8A)^S^.y-x8i.y8^.82.
Similarly y8i . x8^ . /S, = (8^8,) y8i.x- y8, . x8, . S^ .
Also x8i .y + y8i ,x^xy8i + ya?Si = 0.
Therefore xy{8^,8,,8,}^-xy{8,,8,,8,} = xy{8,,8,,8,} (4).
Now let 8, 8\ S" be any three systems of the group G, such that
8^7^y + fi8, + v8,, 8' = \% + fi% + v'8,, 8'' = \'% + fi''8, + v''8s.
Also let A stand for the determinant S ± \fi'p\
Then from equations (1), (2), (3), (4) it follows that
xy[S,8\8''}^Axy{S,,8,,S,},
187] THE CONJUaATB INVARIANT OF A TRIPLE GROUP. 311
This proves the invariant property of xy {8u 8^, S^]. This expression for
the conjugate invariant may now be abandoned in favour of 0 {xy\ in which
the special systems used do not appear.
(5) Let Riy R^, 22, be a conjugate set of systems of the group ; so that
(iJA) = (i2,i2,) = (i2A) = 0.
Then {yU^ . xR^ . R^)^^{yR^ . R^ . xR^^iyR^ . R^ . xRt)
^-{yR^.R^.xRy)^(xR,.yR^.Rt).
Hence 6 (a?y) = i {(a;iJ, . yiZ, . JJ,) + (yiJ^ . ajiZ, . i?,)}
^{xR,.yR,.R,)^{xR,,R,.yR,)^{{xR,.R,)R,.yY
But xRy.Rt.R^^\P(x).
Therefore 0{^)-^h {P ip) y]^\[P iv) ^V
The equation 0(xx)=iO, can be written in the form {P(x)x]^0.
(6) Similarly the condition that the plane X touches the quadnc G is
(X8,.X8,.S,)^0.
The condition that the planes X and Fare conjugate is
{X8,.Y8,.8,) + {Y8,.X8,.8,) = 0.
The expression ^[(X8^.Y8t.8;) + {Y8,.XS^.8^)} can be proved to be
an invariant of the group and will also be called the conjugate invariant,
and denoted by 0{XY). Also
6(ZF) = (?(FZ).
Furthermore if JSi, iZ,, 12, be a set of conjugate systems of the group,
then
Q{XY)^{XR,.YR,.R,).
Also (?(ZF) = i{P(Z)F} = i{P(F)Z},
Therefore the plane-equation of the quadric Q is
{P(J)Z}=0.
(7) Also from § 183 (3), P (x) = JST, (a;) R^ , hence
(? («y) = i [P (w) y} = i [H, (x) R^} = ^ {H, (x) . yR,}
= i{fl-,(yiJ,)«;},from§180(3)
= i{5i(y)J2,4 from §182(1)
= i {H, (y) xR,] = i [H^ (xRO y}.
Similarly G (Z F) = i {iT, (X) YR,} = i {fl^. ( 7R,) X}
= i {JT. ( 7) ZiJ.} = i {H, (XR,) Y}.
(8) Corresponding to each director force A of the triple group, there is
one parabolic subgroup [c£ § 175 (9)]. Let 8^ be any system of this subgroup,
and let H^ {x) and J?, (X) be the harmonic invariants with respect to this
subgroup of a point x and of a plane X. Then we will prove that 5, (x)
[c£ § 179 (7)] is the point of contact of that tangent line from x to the
31 i INVARIANTS OP GROUPS. [CHAP. IH.
quadric, which intersects D^; also that H^iX) is the tangent plane con-
taining that tangent line to the quadric, which lies in the plane X and
intersects the line Di.
For let y be another point, and fif, any sjrstem of the triple group which
does not belong to the given parabolic subgroup. Then, according to sub-
section (2), {{Xx^ fiyhe Sk point on the quadric,
Now let y^Bz (x) = 2xDi . 8^*
Then by § 179 (7), yD, = 0. Hence (yD, , y/Sf^ . ^s) =* 0 ; and
{yD,.x8^,8,)=^0.
Also y82 = 2xDi . 5a . /S^ = a?A (-SA). Hence (xD, . yS^ . S,) = 0.
Thus the equation reduces to V = 0. Hence the two points, in which the
line xHz(x) meets the quadric, coincide at the point H^^x). Similarly for the
second part of the theorem which concerns J7, {X),
Another mode of stating the propositions of this subsection is that, all
quadrics with a given parabolic subgroup touch along the director line of that
subgroup.
Hence the equation of any one of a group of quadrics which touch along
a common generator can be immediately written down.
188. Transformations of 0{pp) and 0(PP). (1) Let a point p
on a plane X be conceived as the null point of X with respsct to some
system of the group; it is proved in sabiection (3) below that, if (?(X, X)
be not zero, p may be any point on the plane X. Hence p can be written
X (Xifii + Xafij + XsiJs); where i2i, JB,, Rj are three reciprocal systems of the
group.
Then writing XqZi?i + XaXiZa + XjXjBs for the second p in 0(pp), and
then using §187 (7)
0 (pp) = \G (p, XR,) + \0(p, XR,) + \,0 {p, XR,)
= i XiJTi {XR,) . ;)22i + i X^H^ (XR,) .pR, + ^ \H, (XR,) . pR,.
But Ri is reciprocal to the dual group g^, and therefore by § 182 (1)
H,{XR,)=-E,{X)R^] similarly E,(XR,)^H,(X)R,, H,(XR,) = H,{X)R, .
Hence
0(pp)=^^\H,{X)R,.pR, + ^\,H,(X)R,,pR,+i^\,H,{X)R,.pR,.
But [cf. § 167, equation (21)]
HAX)R,.pR^=^H,(X)R,.R,.p^i^H,{X)p.{R,R,),
Similarly H^(X)R,.pR,^^H,(X)p .(R,R,l
and Hz(X)R,.pR,=:^Hz(X)p,(R,R,).
Hence
0(pp) = i{MRiRi) Si(X)p + \,(R,R,)H,(X)p + \,(R,R,) H,(X) p] .
188] TRANSFORMATIONS OF Q(pp) AND 0(PP), 313
Again ^H^{X)p^^H,{X),X(\R,-\-\R^ + \Rs).
And by § 187 (7) i Hi (X) .XR,^0 (XX).
Also ^Hi(X).XR, = XR^.It,.XR, = 0; similarly iH,(X).XR,^0.
Therefore ^H,(X)p=^\Q(XX). SimUarly
iJy,(Z)p = X,0(ZZ), ^H,(X)p = \0(XX).
Thus finally 0{pp)^Q (XX) [\' (R,R,) + V (R2R2) + V (-B«Bs)}.
(2) Now if 0 (pp) = 0, |) is a point on the section of the quadric 0
made by the plane X. But O(pp) = 0 involves either (?(XZ) = 0, or
V (liiRi) + V (i2>i2«) + V (^B,) = 0.
If O(XX) = 0, the plane touches the quadric and therefore contains one
director line of the group and one common null line. The null points of
this plane in respect to the various sjrstems of the group must lie on
this common null line.
If Q(XX) be not zero, then
V (iiiiJi) + V (i2A) + V (iisBs) = 0.
But this is the director equation of the group. Hence the director
forces of the group are in general the only systems in respect to which the
null points of any plane — ^not a tangent plane — lie on the quadric.
(3) If G(XX) be zero, then by (2) the three points XR,, XR^, XR^ are
coUinear and lie on the common null line. The point of contact of the plane
is P (X), which is also coUinear with the three points.
If Q(XX) be not zero, the three points XR^^ XR^, XR^ form a triangle
on the plane. Hence any point on the plane can be represented by
X (XiRi + X9/Z, + \RzX that is by XS where 8 is any system of the group.
Also P (X) does not lie on the plane X.
(4) Similarly if P^x(X,R, + \J[t^-\-\,R;),
then 0 (PP) = Q (xx) {V (RiRi) + V (R^R^) + V (RzR$)Y
The plane P is a tangent plane of the quadric 0, if 0 (PP) = 0.
But this equation involves either 0 (xx) = 0,
or V (RiRi) + V (R^) + V (RtRz) = 0.
If 0(^ = 0, the point a? lies on the quadric and therefore is contained in
one director line of the group and one common null line. The null planes
of this point with respect to the various systems of the group all touch the
quadric (since (?(PP) = 0), hence they all contain the common null line
through the point.
If 0 (xx) be not zero, then V (-Bi^) + V (RiR%) + V (R^R%) = 0. But
this is the director equation of the group. Hence the director forces of the
group are in general the only systems of the group with respect to which
the null planes of any point — ^not lying on the quadric — touch the quadric.
314 INVARIANTS OF GROUPS. [CHAP. III.
If 0{xx) be zero, the three planes xRi, a?iJj, xR^ are coUinear, and con-
tain the common null line through x. The tangent plane at ^ is P {x) and
also contains this null line.
If 0 (xx) be not zero, the three planes xR^, xR^, xR^ are not coUinear.
Hence any plane through x can be represented in the form
X (XiJBi + XjJBa + \R^'
Also P (x) does not contain the point x.
(5) Let p, Ply PifPs be the corners of a self-conjugate tetrahedron of a
quadric ; and let a conjugate set of systems of one of the two triple groups
defined by the quadric be
Ri=PPi + fJl'iP2Pa> Ri'^PPi + fJ^PsPu R$=PPB + fhPlP2'
Also let any point x be defined by ^ -f- ^ipi -h fai^a + SsPz-
Then xR^ = fh^pp^p^ + f^^iPiP^Pz + ^iPPiPz + SiPPiP$>
and a?i2j = - fJ^ppiPs - ^iPPiPi + H^i%PiP%Pz + izPP-iPz •
Hence xR^.R^^{ PPiP^Pz) {f^fh^Pi - ^iP + Mr^^Pz - A^f »?«}.
Therefore finally 0 (xx) = (ppiPiPzY {Ati/^aAb? + Ahfi' + f^fa' + /^jfs'}.
Accordingly the equation of the quadric, 0 {xx) = 0, can be written in
the form
f^fh fhfh. /*i/*a
(6) Conversely let the equation,
be the equation of the quadric; where x is the point fy> + fipi + ^^Pi + ^sPs-
Also let one group of the two reciprocal groups, defined by this quadric,
be defined by the conjugate set
PPi + fHP%Pz, PP% + tHPzPu PPi + fhPiP2*
Then by comparison with (5) we find
« _«! «.?2 _?8
Hence ^^-ifi??«l.
Therefore /^/i,/t, = ± . /r^ •
Finally Ah = ± X^«i, fh—± XaOa, and /*, = ± Xaas ;
where all the upper signs are to be chosen or all the lower signs, and 1/X is
put for + i\/(aaia^8).
188] TRANSFORMATIONS OF 0{pp) AND 0 (PP). 315
Thus one group associated with the quadric is defined by the three
systems
The reciprocal group is defined by the three systems
ppi-T^diPtPty pPi-yy^^oi^PtPu PPi-'^^c^PiP*
If aotiOtsa, be positive, X is real; if negative, X is imaginary.
CHAPTER IV.
Matrices and Forces.
189. Linear Transformations in Three Dimensions. (1) Let a
real linear trausformation of elements in a complete region of three dimen-
sions be denoted by the matrix ^, as in Book IV, Chapter VL All the
matrices considered will be assumed to be of zero nullity [cf. § 144 (2)].
(2) The following notation respecting latent and semi-latent regions,
which agrees with and extends that in ^ 145 to 150, will be used.
The region, such that for each point a? in it if>x=ya!, is called the latent
region corresponding to the latent root 7 of the matrix. Any subregion of
this latent region will be called a latent subregion corresponding to the
latent root 7.
(3) The region, such that for each point a? in it ^ = 7a? + y, where y is
a point in an included latent region, is called a semi-latent region of the
first species. Here two cases arise. Firstly, if 7 be a repeated root, then
[cf. § 149] there may be a semi-latent region of the first species corresponding
to the root 7. Similarly [cf. § 150], there may be semi-latent subregions of
species higher than the first.
Secondly, let 71 and 7, be two latent roots, of which either or both may
be repeated ; and let 61,63 be a pair of latent points belonging to the two
roots respectively.
Then [c£ § 146] any point x = ^161 + ^^ is transformed according to
the rule,
^ = 7if 1^ + 7af a^» = 7ia? + fiszeg = 7aa? + SjCi .
The region defined by the assemblage of independent latent points
corresponding to both 71 and 72, that is by the points Ci, 6i\ 6i'\ etc.,
62, e,', 62", etc., is called the semi-latent region of the first species corre-
sponding to the two roots conjointly. If neither of the roots be repeated,
such a region is necessarily a straight line.
(4) A subregion of a semi-latent region of the ath species, which is not
contained in a semi-latent region of a lower species, and which is such that
189, 190] LINEAR TRANSFORMATIONS IN THREE DIMENSIONS. 317
any point x in it is transformed into a point in the same subregion, is called
a semi-latent subregion of the ath species.
The semi-latent regions, or subregions, which are of most importance in
the present investigation, are straight lines. A semi^latent straight line is
necessarily of the first species ; so that its species need not be mentioned.
It . necessarily contains at least one latent point : let ^i be a latent point on
such a line, and x any point on it.
Then the transformation of x takes place according to the law
If 72 be not the latent root corresponding to 6i, then another latent
point €2 exists on the line, and the line is a semi-latent subregion with
respect to the two latent roots 71 and 7a conjointly.
If 72 be equal to 71, the latent root corresponding to e^ then, either
Si = 0, and all the points on the line are latent, and the line is a latent
region (or subregion) ; or Si is not zero, and there is only one latent point
on the line. Thus, if 72=71, the line is either a latent or semi-latent
subregion corresponding to the root 71.
(5) A semi-latent plane is at most of the second species. The trans-
formation takes place according to the law
^zsyx + x' \
where x is any point on the plane, and x' is some point (depending in general
on x) on a semi- latent line lying in the plane.
If the plane be semi-latent of the first species, it necessarily contains lines
which are semi-latent subregions of the first species. For let x\ above, be a
latent point, so that tf>af = 7a?', then the point \x + fix' is transformed accord-
ing to the law
0 (X« + fuxf)^\ (yx + a/) + fji^af^'k/yx + (\ + fJi^)x^»
Hence any point on the line xaf is transformed into another point on the
line xa/ ; and therefore the line is a semi-latent subregion.
190. Enumeration of Types of Latent and Semi-Latent Regions*.
(1) Let the four latent roots 71, 72, y^t 74> of the matrix (f) be distinct.
Then [cf. § 145 (5), (6), (7)] there are four and only four independent latent
points, one corresponding to each root. Let these points be Ci, 69, e^, e^.
The only semi-latent regions of the first species are the six edges of the
tetrahedron 61^2^4; and each edge is semi-latent with respect to two ix)ots
* This ennmeration has, I find, been made by H. Grassmann (the younger) in a note to
^ 877—390 of the Avsdehnungslekre von 1S62 in the new edition, edited by F. Engel (of. note at
end of this chapter). He gives interesting applications to Enclidean Space. He also refers to
Von Staudt, Beitrdg^ zur Geometrie der Lage, 8rd Ed., 1860, for a similar enumeration made by
different methods.
318 MATRICES AND FOBCES. [CHAP. IT.
conjointly. The only semi-latent regions of the second species are the four
faces of the tetrahedron eie^s^^; and each face is semi-latent with respect to
three roots conjointly. All the manifold necessarily belongs to a semi-latent
region of the third species ; therefore such species need not be further con-
sidered.
The enumeration will be made without regard to the difference of type
which arises according as the roots are real or imaginary,
(2) Let there be three distinct latent roots of the matrix. Let these
roots be 71, 7j, 74; and let the root 71 be the repeated root.
There are necessarily [cf. § 148 (2)] three latent points ^, ^, 64, corre-
sponding respectively to the three roots. There is a line ei€^ which is, either
(Case I.) a latent region corresponding to the root 71, or (Case 11.) a semi-
latent region corresponding to the root 71.
Cdae L The line ^i^a is a latent region corresponding to the root 71, ^ is
the latent point corresponding to 7,, and €4 is the latent point corresponding
to 74. The semi-latent regions of the first species are, the plane 616^
corresponding to the roots 71 and 7, conjointly, the plane 616^4 corresponding
to the roots 71 and 74 conjointly, the line 6^4 corresponding to the roots 7,
and 74 conjointly. Any line through «,, which intersects eiC^, is a semi-latent
subregion corresponding to the roots 71 and 73 conjointly ; any line through
€4, which intersects eie^, is a semi-latent subregion corresponding to the two
roots 7i and 74 conjointly.
The complete manifold forms a semi-latent region of the second species
corresponding to all the roots conjointly. All planes through the line 6^4
are semi-latent subregions of the second species corresponding to all the
roots conjointly.
Case II. The point ei is the sole latent point corresponding to the root
7i, the points ^, and €4 are the latent points corresponding to the roots 7, and
74. The semi-latent regions of the first species are, the line e^ oorre-
sponding to the root 71, the line eie^ corresponding to the roots 7, and
7s conjointly, the line 6^64 corresponding to the roots 71 and 74 con-
jointly, the line e^4 corresponding to the roots 73 and 74 conjointly. The
semi-latent regions of the second species are the planes ^i^a^s, 616264, 61^4;
the roots to which they correspond need not be mentioned.
(3) Let the matrix have a triple latent root 71 ; and let 74 be the other
root.
There are necessarily two latent points e^ and 64 corresponding to these
roots respectively. There is [cf. ^ 149, 150] a plane eie^ which is, either
(Case I.) a latent region corresponding to the root 71, or (Case II.) a semi-
latent region of the first species corresponding to the root 71, or (Case III.)
a semi-latent region of the second species corresponding to the root 71.
190] ENUMERATION OF TYPES OF LATENT AND SEMI-LATENT REGIONS. 819
According to § 150 (5), the points e^ e^, e^ can always be so chosen that
where 82 cannot vanish unless Si also vanishes.
Thus, in Case I. Si = 0 = S, ; in Case II. Si = 0, and S, is not zero ; in Case
III. neither Si nor Sg vanishes.
Any point ^ = S^^ is transformed according to the rule
^ = (7if 1 + ^if 2) ei -f (7if , + Sjf 3) $2 + 7if ««8 + 74?^.
• Thus in Case L, ^=7i« + (74--7i)f4«4 (A).
In Case II., <A^ = 7i«+S2f«^ + (74-7i)f4^4 (B).
In Case III., <^ = 71^? + Sif j^ + S^f s^, + (74 - 7i) f 4^4 (C;.
Case I. The plane «]^s is the latent region corresponding to the triple
root 7i, and 64 is the latent point corresponding to the root 74. Any line
in the plane CiB^ is a latent subregion corresponding to the root 71.
The complete manifold is the semi-latent region of the first species
corresponding to the two roots conjointly. Any line through ^4 is a semi-
latent subregion corresponding to the two roots conjointly.
Case II. The latent regions are, the line ^16, corresponding to the root
7i, and the point 64 corresponding to the root 74. The semi-latent regions of
the first species are, the plane Sie^ corresponding to the root 71, and the
plane ei^sei. There is some one point [cf. § 149 and § 150 (5)] in the line eie^
(the point e^, according to the present notation), such that any line through
it in the plane eie^ is a semi-latent subregion corresponding to the root 71.
Also any line through €4 in the plane eiC^t is a semi-latent subregion corre-
sponding to the roots 71 and 74 conjointly. The semi-latent region of the
second species is the complete manifold. Any plane through e^4 is a
semi-latent subregion of the second species ; for from equation (B)
e^4<f)x = e^^.
Case III, The only latent points are, the point Ci corresponding to the
root 7i, and the point €4 corresponding to the root 74. The semi-latent
regions of the first species are, the line eie^ corresponding to the root 71, and
the line ei«4 corresponding to the roots 71 and 74 conjointly. The semi-latent
regions of the second species are the planes eie^, and the planes eie^4.
(4) Let there be only one latent root to the matrix, which occurs
quadruply. Let 71 be this root, and let ei be the latent point corresponding
to it which necessarily exists [cf. § 148 (2)]. Then [cf. § 150 (5)] it is always
possible to find three other points e^, e^, 64, such that
^4 = 7i^4 + S,tf„ ^ = 7i^+S^,, ^=7A + Si6i, ^ = 71^1.
Then any point a 3= X^e is transformed according to the rule
<^ = (7if 1 + ^if «) ^1 + (7if « + Sgf s) «2 + (7if s + §»f 4) et + 71^4^4
= 7ia^ + 8if^i + S»f.e.+ ?.f4^ (I>X
1
320 MATRICES AND FORCES. [CHAP. IV.
live cases now ai*ise according as, either (Case I.) £,, S, and S, all vanish ;
or (Case 11.) Bi and S2 vanish, but S, does not vanish ; or (Case III.) Bi
vanishes, but S^ and Sj do not vanish ; or (Case IV.) Sj, ^a> S3 do not vanish ;
or (Case Y.) B^ vanishes, but B^ and B^ do not vanish.
Thus in Case L, ^ — jiX ^ (E).
In Case IL, ^x^^x^+B^^^ (F).
In Case III., ^ = 7i« + Saf s«a + ^sf A (G)-
In Case IV., ^x=^r^xX^\^ByJ^^^ + Bj^^ + BJ^^e^ (H).
In Case v., </wc == 71^ + Sif a^i + S,f4^ (I).
Case I. Every point is latent ; and the operation of the matrix is simply
equivalent to a numerical multiplier. This case need not be further con-
sidered ; and may generally be neglected in subsequent discussions^
Cdse II. The latent region is the plane 616^^ The semi-latent region
of the first species is the complete manifold. Every line through the latent
point 63 is a semi-latent subregion. All semi-latent planes must pass
through Ct ; and all planes through e^ are semi-latent.
I
Case IIL The latent region is the line ^eg. The semi-latent region of
the first species is the plane eie^. Every line through e^ and lying in the
plane €16^ is a semi-latent subregion. The semi-latent region of the second
species is the complete manifold. Every plane through the line e^ is a
semi-latent subregion of the second species ; for by equation (G)
e^<f>x = e^s^.
Case IV. The sole latent point is ei. The semi-latent region of the first
species is the line eiS^. The semi-latent region of the second species is the
plane eie^. The semi-latent region of the third species is the complete
manifold.
Case V. The latent region is the line eie^. The semi-latent region of
the first species is the complete manifold. Every point lies on some straight
line which is a semi-latent subregion. For consider the condition that the
point oc{=^^e) may lie on a semi-latent straight line through the latent
point a (= ttiei + Oj^).
Now if>x = jiX -f- 8,f i^i + 88^4^ = 7i^ + ^CL> by hypothesis.
Hence ^fa = ^i, 8,^4 = Xat.
Thus X must lie on the plane of which the equation is
This plane passes through a ; and all lines lying in it which pass through
a are semi-latent. A planar element in this plane is ei^(ai£8^3 + oitBiei).
(5) Let thei*e be two distinct latent roots, and let each be once repeated.
Let 7i and 73 be the two distinct latent roots ; and let ei and e^ be the two
latent points, which certainly exist, corresponding to these roots respectively.
190] ENUMERATION OF TYPES OF LATENT AND SEMI-LATENT REGIONS. 821
Then it is always possible to find two points e, and e^, such that
Hence any point x = Sfe is transformed according to the rale
^ = (7lf 1 + Slf j) ei + 7if gCa + (78^, + 83^4) ^8 + 7«f 4^4.
Three cases now arise according as, either (Case I.) Bi and 83 both vanish ;
or (Case II.) Si vanishes and Sg does not vanish ; or (Case III.) neither Si nor
S, vanishes.
The case, when S, vanishes and Si does not vanish, is not a type of case
distinct firom Case IL
Thus in Case I.,
<^ = 71^7 + (7, - 7i) (f ,63 + f 4^4) = 7ta? + (71 - 7»)(fi^ + f«^) (J)-
In Case II.,
<A^ = 7i^ + {(7« ~7i) ft + 8»f4) ^ + (7» - 7i) ^4^4
= 78«^+ (71 - 73) (fl^l + ^^) + S»f4e8 (K).
In Case III.,
<A^ = (7lfl + *if 9) «1 + 7lf2«a + (78f8 + 8,^4) ^S + 7,^4^4 (L).
Oflwe /. The latent regions are, the line Cie^ corresponding to the root 71,
and the line e^^ corresponding to the root 7,. The semi-latent region of the
first species is the complete manifold, and it corresponds to the two roots
conjointly. Every line intersecting both €162 and e^^ is a semi-latent sub-
region corresponding to the two roots conjointly.
Case IL The latent regions are, the line eie^ corresponding to the root
7i, and the point e^ corresponding to the root 73. The semi-latent regions
of the first species are, the line ^3^4 corresponding to the root 73, and the
plane eie^ corresponding to the roots 71 and 73 conjointly. Every line
through es in this plane is a semi-latent subregion corresponding to the two
roots conjointly. The semi-latent region of the second species is the
complete manifold. Every plane through ^4 is a semi-latent subregion.
Case III. The only two latent points are, ei corresponding to the root 71,
and eg corresponding to the root 73.
The semi-latent regions of the first species are, the line Ci^a corresponding
to the root 71, the line e^4 corresponding to the root 73, the line 61^3 corre-
sponding to the roots 71 and 73 conjointly. The semi-latent regions of the
second species are, the plane eie^ corresponding to the roots 71, 71, 73 con-
jointly, and the plane eie^e^ corresponding to the roots 71, 73, 73 conjointly.
It is to be noticed that, in order to define the roots corresponding to the
semi-latent regions of the second species, the repeated roots must be counted
twice. The semi-latent region of the third species is the complete manifold.
w. 21
322 MATRICES AND FORCES. [CHAP. IV.
191. Matrices and Forces. (1) Let S denote any system of forces,
and ^ any matrix. Then ipS is defined in § 141 (1) and (3), and denotes
another system of forces.
If ^fif = S, the system S is said to be a latent system of the matrix (f)
[cf. § 160 (1)].
If every system Sf, belonging to a group of systems G, is transformed
into another system <f)S of the same group, the group 0 is said to be a semi-
latent group of the matrix.
If every system of the group is latent, the group is said to be latent.
(2) The following properties of the transformation are immediately
evident.
If (S8') = 0, then {<f>8<f>S') = (</» . SS") == 0. Hence if S and S' are reciprocal,
<I>S and <f>S^ are also reciprocal ; and if S reduce to a single force, <^/S reduces
to a single force.
If fif belong to the group defined by fif,, S„ ... Sp, then if>S belongs to the
group <f>Si, <l>Siy ... ^/Sfp.
(3) Let BiB^^^ be any fundamental tetrahedron of reference; and let
and if)S = -oTu'eiej + vr^^e^ + ^^e^x + 'Bri/^^4 + '^u^i + ^u^4*
Then it is obvious [cf § 141] that the coeiSScients istu, ..., vr^ are transformed
into the coefficieuts w^\ ..., w^' by a linear transformation represented by a
matrix of the sixth order.
But the most general matrix of the sixth order contains thirty-six
constants ; whereas the matrix of the fourth order, from which the present
transformation is derived, contains only sixteen constants. Accordingly
relations must hold between the thirty-six constants reducing the number
of independent constants to sixteen.
(4) An interpretation of these relations can be found as follows. In
general the transformation of the sixth order yields only six latent systems :
and in general a matrix of the fourth order has four latent points forming a
tetrahedron. The six edges of this tetrahedron are latent forces. Hence in
general the six latent systems are six single forces along the edges of a
tetrahedron. The expression of these conditions yields twenty independent
equations, which reduce the number of independent constants to sixteen.
(5) Let the comers of the tetrahedron €16^8^4, be the latent points of the
matrix 0. This is always possible, when the roots 71, 72, 7,, 74 are unequal
Then the latent roots of the matrix of the sixth order, which transforms
the co-ordinates of any system of forces, are given by the sextic
(o- - 7i7») (o- - 7,74) (a - 7^s) (o- - 7174) (a- - 7^71) (cr - 7^4) = 0.
191, 192] MATRICES AND FORCES. 323
(6) Apart from the special cases when some of the roots of the matrix ^
are repeated, the only cases, for which latent systems exist other than forces
along the six edges of the tetrahedron, arise when two roots of this sextic are
equal [cf. § 145 (4) and § 148], There are two types of such equality;
namely, the type given by 7i72 = 7i0'4, and the type given by 7^3 = 7,^1.
The second type necessitates 71 ~ 72 ; and thus leads back to the special cases
when the matrix <f> possesses repeated latent roots.
(7) It is evident from (3) that, if Si, S^, ... /?« be any six independent
systems, and any system S be defined by
S = XiSi + \^i + . . . + XflSe,
and <f>S = X/Si + X/^Sa + . . . + VS„
then Xi\ V> ••• V> can be derived from Xj, Xa> ••• \j hy a linear transformation.
192. Latent Systems and Semi-Latent Groups. (1) A force on
any semi-latent line is a latent force. For let ^le, be the semi-latent line,
and 6i the latent point on it. Then [c£ § 189 (4)] we may assume,
Hence ^^lea = <^^^ = 7172^1^2.
(2) If one of two conjugate forces of a latent system be latent, the other
force is also latent. For let /Sf = Xi)i + /*A; and assume that (f>S=S,
^i)i = A- Then since
(1)8 = x</»A + /^</>A = xA + /*A,
it follows that ^A = A«
Also from subsection (1), ^A = 77'A, and 0A = 7VA; where 7, 7',
7'', y'" are latent roots of the matrix, but not necessarily distinct roots.
Thus yr/ = 7 V"- This forms another proof of § 191 (6).
(3) Hence, if a latent system S (not a single force) exist and also
a semi-latent line which is not a null line of 8, then another semi-latent
line not intersecting the first must exist, and such that the two lines are
conjugates with respect to 8.
(4) The null plane of a latent point with respect to a latent system
is semi-latent.
For let e be the latent point, 8 the latent system, P a planar element in
the null plane. Then
P = Xcfif, 4>P^\^8 = \<l>e<l>8^e8 = P,
Conversely the null point of a semi-latent plane with respect to a latent
system is latent.
(5) Hence from (2), (3), (4) it is easy to prove that, if a tetrahedron of
latent points exist, every latent system must have two semi-latent lines as
one pair of conjugates.
21—2
324 MATRICES AND FORCES. [CHAP. IV.
(6) If a group is semi-latent, its reciprocal group obviously is also
semi-latent.
(7) In general a semi-latent group of /> — 1 dimensions (p ^ 6) contains
p latent systems. This follows from § 191 (3) and (7).
(8) If no special relation holds between the latent roots of the matrix,
then [cf. § 191 (4)] the only six latent systems are six single forces on the six
edges of the tetrahedron formed by the four latent points. It follows that
in general a semi-latent group of (/o — 1) dimensions must have p edges of
the fundamental tetrahedron of the matrix as director lines.
Thus in general a dual group of the general type can be a semi-latent
group, namely, any dual group with two non-intersecting edges of the
fundamental tetrahedron as director lines. Also [c£ subsection (6)] in
general a quadruple group of the general type can be a semi-latent group,
namely, any quadruple group with two non-intersecting edges of the funda-
mental tetrahedron as common null lines.
But in general a triple group of the general type cannot be semi- latent.
For the director lines of a triple group, which are generating lines of the
same species of a quadric, do not intersect unless the quadric degenerate into
a cone or into two planes. But there are not three non-intersecting edges
of a tetrahedron.
(9) A matrix can always be constructed with four assigned latent roots,
so that any given dual group (or any given quadruple group) is semi-latent.
For it is only necessary to choose two of the latent points on one director
line (or common null line), and two on the other director line (or common
null line).
(10) Every semi-latent dual group, which is not parabolic and does not
consist of single forces, contains at least two distinct latent systems, which
are either the director forces or two reciprocal systems.
For, if Di and D, be the director forces of the group, then JDi and D,
remain the director forces of the group after transformation. Hence either
ipDi = Di, and ^D, = A ; or 0A = A, and ^A = A-
In the first case the two director forces are the two latent systems.
In the second case, let if>Di^oJ)^, ^A = )3A. Let one of the latent
systems be S=^7lDi + fiD^.
Then 0iS = <rfif = X^ A + /*</» A = XoDa + M'fiDi-
Hence a\ = fi^, a-fi=\cu
Therefore a* = a/9.
Hence — = + -l_
V/8 -Va'
192] LATENT SYSTEMS AND SEMI-LATENT GROUPS. 325
Thus the two latent systems are
VySA + Vai), and V/SA - 'JoJ)^]
and these systems are reciprocal [cf. § 173 (6)].
But reciprocal systems of a dual group of the general type are necessarily
distinct.
(11) The director force of a semi-latent parabolic group is evidently
latent.
Also [cf. § 172 (9)] the null plane of any point on the director line is the
same for each system of the. group. Hence the null plane of every latent
point on the director line is semi-latent. Thus there must be as many semi-
latent planes passing through the director line as there are null points on it,
for a semi-latent parabolic group to be possible. And conversely there must
be as many latent points on the director line as there are semi-latent planes
through it.
(12) If a semi-latent line exists which does not intersect the director
line of a semi-latent parabolic group, then the parabolic group contains at
least one latent system in addition to its director force.
For let D be its director force and JV" the semi-latent line. Then [cf.
§ 172 (10)] one and only one system of the group exists, for which N is
a null line. This system is D (N8) — S (ND) ; where S is any other system
of the group. But since the group is semi-latent, and the lines N and D
are semi-latent, this system must be latent.
(13) Let D = 6i«3, and N=^e^^; and let Ci and «, be the latent points on
D and JV" which certainly exist. Then 61^8 is a semi-latent line. Now by
subsections (3) and (4), either eie^ is not a null line of the latent system, and
then its conjugate is also a latent line intersecting both D and N in two
other null points ; or eie^ is a null line of the system.
In this last case e^ is the null point with respect to the latent system of
the plane eiN, since the two null lines eie^ and JV intersect in it: also ei is the
null point of the plane ej) for all the systems of the group [cf. § 172 (9)],
since the null lines eie^ and D intersect in it.
(14) It follows from (12) that the only possibility, for the existence of a
semi-latent dual group with only one latent system, is when all the semi-
latent lines intersect one of their number. Then such a semi-latent group
is parabolic, and the director force is the single latent system, and is on that
one of the semi-latent lines intersected by all the rest.
Furthermore, if two non-intersecting semi-latent lines exist, and no latent
system exists which is not a single force, then no semi-latent parabolic group
with either of these lines as dii*ector line is possible. For by (12), such a
group must contain a second latent system, and by hypothesis such a system
does not exist.
326 MATRICES AND FORCES. [CHAP. IV.
(15) Every semi-latent triple group of the general type must contain
at least three distinct independent latent systems, unless it contains a
semi-latent dual group with only one latent system. This is obvious,
remembering that the properties of semi-latent triple groups are particular
cases of the properties of the linear transformation of points in a complete
region of two dimensions [cf. § 145 (4) and § 148].
Also by the preceding subsections such a semi-latent dual group is
parabolic. Hence, if for any matrix no semi-latent parabolic group with
only one latent system exists, every semi-latent triple group must have three
distinct latent systems.
(16) If a plane region of latent points exists, every director force D
of a semi-latent triple group intersects this plane, and therefore has a latent
point on it. Thus D and ^D are either congruent or intersect. If they
intersect, the triple group is not of the general type, since <^i) is also a
director force. Hence the only possible type of semi-latent triple group
is latent.
(17) If a line of latent points exists, this line is either a generator of
any quadric or intersects it in two points. Now consider the quadric
defined by any semi-latent triple group 0. The reciprocal group Q' is also
semi-latent.
Firstly let the latent line intersect the quadric. Then by the same
reasoning as in the previous subsection (16), two director forces of G and
two of (?' must be latent, assuming that 0 is of the general type.
Secondly let the latent line be a director line of G. Then as in (16) all
the director forces of 0^ (null lines of G) are latent.
Thirdly let the latent line be a null line of G\ Then all the director
forces of G are latent.
193. Enuaieratign of Types of Latent Systems and Semi-Latent
Groups. (1) Let no two roots of the sextic of § 191 (5) be equal. The
four latent roots of the matrix ^ are unequal, and only four latent points
«i» «2, ^zy ^4 exist. Then [cf. § 191 (6)] the only latent systems are the six
forces on the edges of the tetrahedron e^e^^^.
By § 192 (10) the only semi-latent dual groups, not parabolic, have two
edges of the tetrahedron as director lines.
No semi-latent parabolic group can exist, for by § 192 (12) and (14) a
latent system of the group must exist which is not a director line ; and there
is no such system.
No semi-latent triple group of the general type can exist. For [c£ § 192
(15)1 such a group must contain three latent systems, that is, three edges
of the tetrahedron eie^s^^ as director lines. But three non-intersecting edges
cannot be found.
193} ENUMERATION OF LATENT SYSTEMS AND SEMI-LATENT GROUPS. 327
The semi-latent quadraple and quintuple groups can be found by the
use of § 192 (6). Thus it is not necessary to enumerate them.
(2) Let the four roots 71, 72, 7,, 74 of the matrix be unequal, and [cf.
§191 (6)] let 7x78=7:^4.
Case I. Let no other roots of the sextic of § 191 (5) be equal. Then, as
in (1), the four latent points €16^8^4 form a tetrahedron. The latent systems
are the six single forces on the edges of the tetrahedron, and any system of
the type Xci^ -f- fis^^. It can be seen by the use of § 192 (2) that no other
system can be latent. The dual group defined by ^i^, and e^^ is therefore
latent. The dual groups defined by any system of the type X^e5 + /A«a^4
together with ei^ai or ^4, or e^e^y or e^ are semi-latent. They are parabolic
groups. The only semi-latent dual groups, not parabolic, have two edges of
the tetrahedron as director lines.
By § 192 (14) no semi-latent dual group exists with only one latent system.
The semi-latent triple groups of the general type are, all groups of the
tjrpe defined by Xc,6a + /xeae4, 61^8,^4; and all groups of the type defined by
The generality of these types is proved by § 175 (9) and (12). By
§ 192 (15) no other semi-latent triple group (of the general type) exists.
Case II. Let 717, = 7^4, and 7174 = r^^y^. Then 71' = 72*, 7,* = 74*. Hence,
excluding the case of equal roots which is discussed later [in subsection (5)],
7i=-7a> 78= -74.
As in Case L there are only four latent points, which form a tetrahedron.
The latent systems are, the six single forces on the edges of the tetrahedron,
and [c£ § 191 (6)] any system of the type Xeies + /i«2&4, and any system of
the type \eie^ -h fjLe^,
The semi-latent dual groups are, the semi-latent group defined hy e^e^
and ^,64, the latent group defined by eie, and e^e^y the latent group defined
by ^164 and e^e^^ any group defined by systems of the types X^i6^ + /i^&4 and
^'^1^4 + At'^jg,. This last tjrpe of semi-latent group is not parabolic, unless
one of the four quantities \ fi, \\ /a' vanishes.
The semi-latent triple groups of the general type are all groups defined
by sets of three systems of the following types :
^€», e^4, >A^ + At«i^4,
^«2, ^4, M«4 + /A«a^,
6164, e^y X6i^ + AtV4,
ei^s, 62^4, \eie4 + fie^.
Thus there are four types of semi-latent triple groups for this case.
There appears to be a fifth type of group of which a typical specimen is
the group defined by Xeie» + fie^*^ X'eiCt -h fie^, 616,. But this group cannot
328 MATRICES AND FORCES. [CHAP. IV.
be of the general type. For from § 175 (9) only one parabolic subgroup of a
triple group (of the general tjrpe) exists with a given director line of the
triple group as its director line. Whereas in the group above mentioned,
ei^, is the director line of two such subgroups.
By § 192 (15) no other semi-latent triple group of the general type
can exist.
(3) Let 7i = 72. Then from § 190 (2) there are two cases to be con-
sidered ; but two extra cases arise, when the relation 71' = 7,74 is satisfied.
Thus there are four cases in all ; in the first two cases it is assumed that
7i' 4= 7*74- Let the notation of § 190 (2) be adopted.
Case /, [Cf § 190 (2), Case I.] The line eie^ is the latent region corre-
sponding to the root 71. Then from § 190 (2) the latent systems are, ejej, 6^64,
any force of the type e^ (XCj + /a^j), any force of the type e^ (Xa + H^-
From § 192 (3) no other latent systems exist. Hence no latent systems
exist, which are not single forces.
The semi-latent dual groups (not consisting entirely of single forces)
are the group defined by 616^^ e^^] and the groups of the type defined by
No semi-latent parabolic group exists [cf. § 192 (14)].
No semi-latent triple group of the general type exists ; since three non-
intersecting latent forces do not exist, and the only latent systems are single
forces.
Case IL [Cf § 190 (2), Case IL] The line eie^ is the semi-latent region
corresponding to the root 71. The latent systems are the forces eie^^ 616,,
€164, 6^4. It is impossible for any latent system (not a single force) to
exist, since ei^s, ^i^,, €164 are not coplanar, and cannot therefore all be null
lines. Hence the theorem of § 192 (3) applies ; and the truth of the state-
ment can easily be seen, since Xei^a -h fie^i is not latent.
The only semi-latent dual group, not consisting entirely of single forces,
is that defined by ^i^,, 6^4,
No semi-latent parabolic group can exist [cf. § 192 (11) and (14)].
No semi-latent triple group of the general type can exist [cf § 192 (15)}
Case III, (Subcase of Case L) Let 71* = 7,74 ; and let the arrangement
of latent and semi-latent points and regions be that of Case I. Then, from
Case I. and § 192 (3) the only latent systems are, ei^g, ^4, any force of the
type ^ (X^i + ftea), any force of the type €4 {Xsi + fcea), and any system of the
type X^ea + fie^^'
The semi-latent dual groups (not consisting entirely of single forces) are
the (latent) group defined by Cie^, ^4; any group of the type defined by
«» 0^ + H^)> ^4 (^'^1 + A*'^) > ^^^ parabolic groups of the types defined by
193] ENUMERATION OF LATENT SYSTEMS AND SEMI-LATENT GROUPS. 329
X^«a + iie^i, together with either e^ (V^ + /x'^a), or e^ (X'^ + ii'e^. No other
semi-latent parabolic groups exist [cf. § 192 (14)]. The semi-latent triple
groups are groups of the type defined by
Case IV. (Subcase of Case 11.) Let 71* = 7,74 ; and let the arrangement
of latent and semi-latent points and regions be that of Case II. Then the
latent systems are the forces ^i^a, 616^, e^e^, e^^, and any system of the type
X^i^j + /i^4. By § 192 (3) no other latent systems exist.
The semi-latent dual groups, not consisting entirely of single forces, are,
the (latent) group defined by ei^a, e^^* all parabolic groups of the types
defined by \aie^-¥ fie^^ together with ^^ or ^164. No other semi-latent
parabolic groups can exist [cf. § 192 (11) and (14)]. No semi-latent triple
group of the general type can exist.
(4) Let 7i=7a=7,. Then from § 190 (3) there are three cases to be
considered.
Case /. [Cf. § 190 (3), Case L] The plane e^e^ is the latent region
corresponding to the root 71. The latent systems are, any force in the plane
€16^1, any force through the point e^.
No other latent system exists. For all the latent lines in the plane ^e^
cannot be null lines, since all the null lines lying in a plane must pass
through the null point. Thus some line he lying in the plane ^i^^, may be
assumed not to be a null line of any such latent system. Then by § 192 (3)
the system must be of the form \e^a + /Lt&o, where 6 and c lie in the plane
^i> ^9> ^> aiid ^4^ ^ ^^y lii^6 through e^. Let a be assumed to be the point
in which this line meets the plane e^e^.
Hence ^ (}s£^a + fihc) = 7i74X^4a -h yi^/jJ>c,
Therefore such systems are not latent.
The semi-latent dual groups are all groups of the type defined by e^a and
6c, where be lies in the plane €16^.
No semi-latent parabolic group can exist [cf § 192 (14)].
No semi-latent triple group of the general type can exist [cf § 192 (16)].
Case II. [Cf. § 190 (3), Case IL] The line ^jCa is the latent region
corresponding to the root 71; and the plane eie^ is the semi-latent region
of the first species corresponding to the first root. Also ea and e^ are such
that ^68 = 7163 + 89^, where S, is not zero. The latent forces are, ^i^j, any
force through e^ in the plane eie^, any force through e^ in the plane eie^^.
No other latent systems exist. For a similar proof to that in Case I. shows that
a latent system, not a single force, cannot have two non-intersecting semi-
latent lines as conjugate lines. Hence such a system must have all the semi-
latent lines as null lines. Therefore e^ must be the null point of the plane
330 MATRICES AND FORCES. [CHAF. IV.
eie^, and 64 the null point of the plane €4,6162- But since the null line ^le,
does not go through e^y which is the null point of the plane 616^4, this is
impossible.
The semi-latent dual groups (not entirely consisting of single forces) are
all groups defined by a force through e^ in the plane BiB^ and a force
through 64 in the plane 616^4.
No semi-latent parabolic group exists [cf § 192 (14)].
No semi-latent triple group of the general type exists [c£ § 192 (15)
and (17)].
Case III. [Cf. § 190 (3), Case III.] The only latent points are, the
point e^ corresponding to 71, and the point e^ corresponding to 74.
The only latent systems are, the force eiej, and the force ^1^4. There can
be no other latent system (not a single force). For by § 192 (4) the null
point with respect to such a system of the semi-latent plane 616^^ is ei,
and therefore the null point of the semi-latent plane e^e^^ is e^. Hence the
line 6164 is not a null line of such a system ; and therefore from § 192 (3)
another semi-latent line, not intersecting 6164, is required. But such a line
does not exist.
There are no semi-latent dual groups, not consisting entirely of single
forces. For the only possibility of such a group lies in the possibility of a
semi-latent parabolic group with only its director force latent [cf. § 192 (14)].
But 6164 cannot be the director force of such a group, since by § 192 (11)
there must be as many semi-latent planes containing ^164 as there are latent
points on 6164, But ^ and 64 are both latent points ; while 616^4 is the only
semi-latent plane through 616^4. Again BiB^ cannot be the director force, for
by § 192 (li) there ought to be as many latent points on it as there are
semi-latent planes through it. But there are two semi-latent planes through
it, namely b^b^^ and b^b^4\ and there is only one latent point on it.
There are no semi-latent triple groups of the general tjrpe [c£ § 192 (15)].
(5) Let there be only one latent root 71 of the matrix. Then [cf. § 190
(4)] there are five cases to be considered: but, of these, the first may be
dismissed at once.
Cobb II. [Cf. § 190 (4), Case IL] The latent region is the plane Byfi^^.
The latent systems are of the tjrpe X^^ + /^c, where a is any point and
the force he is any force lying in the plane BiB^.
If S be any one of these latent systems, <f>8 = 71'iS.
The semi-latent dual groups (not consisting entirely of single forces) are,
all groups defined by the forces b^ and 6c, where a is any point and he is any
line lying in the plane BiB^z\ and any parabolic group with a director force
of the type ^, where d lies in the plane BiB^. All the semi-latent dual
groups are thus either latent, or else possess only one latent system.
193] ENUMERATION OF LATENT SYSTEMS AND SEMI-LATENT GROUPS. 331
In order to prove the above statements, first notice that, if Si and S^ be
the latent systems of a semi-latent group with two distinct latent systems,
<f)Si = yi^Si and <l>Si = yi^8<i. Hence the group is latent, and hence the
director forces are latent, if there are two of them.
Accordingly the director forces of a non-parabolic semi-latent group are
of the tjrpe described.
Again [cf § 192 (11)] the number of latent points on the director line of
a semi-latent parabolic group is equal to the number of semi-latent planes
passing through it. Hence a force of the type e^ is the only possible
director force of such a gi-oup [cf § 190 (4), Case II.]. Now if d be any
point on this director line, a system S of one of the parabolic groups with
this line as director line can be written in the form S=\e^ + fidy, where
a? and y are any points [cf. § 172 (9)]. Now the dual group defined by
e^d and S is semi-latent.
For [cf. § 190 (4), equation (F)],
<f>S = \<l>et<l>x -h fi4>d<f>'y
= 7i^ (7i^ + ^) + 7i/*^ (7iy + ^'^)
Thus <f>8 also belongs to the dual group.
Furthermore, if S' be not zero (that is, if dy do not lie in the plane
616^3), e^ is the only latent system of the group. But if 8' be zero (that is,
if dy do lie in the plane CiC^), every system is latent and the group is
therefore latent.
No semi-latent triple group of the general tjrpe exists [cf § 192 (16)].
Case III. [Cf § 190 (4), Case III.] The latent region is the line eie^.
The latent systems are all forces in the plane eie^ through the point e^.
There are no other latent systems : for all the planes through e^ are semi-
latent; and their null points with respect to any latent system must be
latent [c£ § 192 (4)]. But e^ is the only latent point on all these planes.
There are no semi-latent dual non-parabolic groups (not consisting entirely
of single forces). For there are evidently no semi-latent dual groups (not
single forces) with two latent systems. There are semi-latent parabolic
groups of the type defined by e^ and ab H- \ejc ; where a is any point on the
plane eie^, but not on eie^ or e^ [cf § 192 (11)], b is any point on the plane
eie^s, k ia any point, and \ is determined by a certain condition. In order
to prove this, note that the form assumed is obvious from § 192 (11) and
from the consideration, that the null point of the semi-latent plane ^6^
is latent and lies on the director force e^, and must therefore be e^. Now
let a = aie, -h Oaes + Ose,, where neither ai nor a, is zero ; let 6 = fijCi 4- fi^ + /3,^ ;
let k = KiBi H- ic^2 + fc^i + ^A'
382 MATRICES AND FORCES. [CHAP. IV.
Then, using § 190 (4), equation (G),
<f)(ah + Xejc) = 71^ (ab + 'Kejc) + B^i {a^ — ^^b^) + ^^iSsKte^.
Now <f> (ab + \ejc) must belong to the group defined by 06 + 'Kejc and ae^.
Hence
Hence B^ia^e^b — "Krf 18^X4,6^^ = 0.
(It is easy to deduce from this equation another proof of the limitation of
the position of a, namely that it is not to lie on eiea or on e^ez.)
Again e^b = 61^2^ = e^e^ = ae^^a, say.
Then X=^?^.
These semi-latent parabolic groups have only one latent system.
There are no semi-latent triple groups of the general type [cf. § 192 (17)].
Case IV. [Cf § 190 (4), Case IV.] The sole latent point is the point ei-
The only latent system is the force ^iCg.
The semi-latent dual groups are parabolic groups, with ^^ as axis, and
with 61 as the null point of the semi-latent plane eie^. Such semi-latent
groups have only one latent system, namely the director force CiCa- Also all
such groups are not semi-latent ; one condition has to be fulfilled, which
will be investigated as follows.
Let k = KiBi + K^ -h Ac,e, + /C464, where k^ is not zero ; let a = aiSi + cr^,
where a, is not zero; and let b^^iCi + ^tA + ^8^> where /Ss is not zero. Then
a parabolic group of the specified type is defined by eia and ejc + \a6. If k,
a and b are given, then \ is determined by a condition, which is found as
follows.
From § 190 (4), equation (H), remembering that CiCj = eia = e^,
<l> {eJc + Xab) = 71' (eJc + \a6) + f ^a + yiB^^^ez + 'K/yiSjO^b ;
where f need not be calculated.
But (f> (eje + \ab) belongs to the parabolic group.
Thus 7iSs^4^i^8 + ^YiSiOa^ft = €16^.
Hence 7i^8^4^i^8 + ^^a^ejb = 0.
Also 61636 == P^e^ .
Therefore 7i8s/t4 + X^itidifiz = 0.
But 7i, 5i, Os, ^Ss, K^ are not zero.
Hence \ = — ^^ — 3- .
No semi-latent triple group exists. For if such a group existed, e^e^ must
be a director line. But the reciprocal group must also be semi-latent, and
therefore it must also contain 6169 as a director line.
_r ■ -
193] ENUMERATION OF LATENT SYSTEMS AND SEMItLATENT GROUPS. 333
Case F. [C£ § 190 (4), Case V.J The latent region is the line e^e^. Those
latent systems which are single forces are grouped in planes: thus corre-.
spending to the latent point a (= cci^ + Oa^s), there are an infinite number of
latent forces of the tjrpe ax \ where x is any point on the plane of which the
equation is
Any point x lies on the line of a latent force.
For x(f>x = X (Sif jft + S,f46s).
Hence <^ {xif>x) = r^^x (hi^^ + Sjf A) = 7i"«<^.
Also the force 6i6s is latent.
The oDly latent systems, which are not single forces, are formed by any
two latent forces as conjugate forces: thus, if x and y be any points, x^ + yif>y
is a latent system. And if /S be such system ^fif = ^^S. Also it may be noted
that eiB^ is a null line of all such systema
There is no d priori impossibility in the existence of latent systems with
all the semi-latent lines as null lines. The following investigation shows
that such a latent system does not exist.
For, since 61^8, 61^4, e^ are to be null lines, systems, with all the semi-
latent lines as null lines, must be of the form
This form is found by assuming
then the following equations must hold
Also any point a (= tti^i + Ojes) on the Une 61^3 is the null point of the
semi-latent plane eie^ipL\h^-{' afi^e^y for all values of the ratio of ai to as.
Hence e^e^ (a^S^ + 0,81^4) . (f i«ei6, + ^uBiB^ + t^Afiz) s ftiei + 0,68 ;
that is tti^sf i4«i + aJ^^T^ = aA + Os^s-
Hence systems with all the semi-latent lines as null lines must be of
the type
But by operating on such a system with the matrix, it is easy to see
that such a system is not latent. For, if ^ be the system,
<^iSf = 7i»fif + 2|7ASA^.
Any dual group defined by two latent systems is latent, and has therefore
(unless it be parabolic) two latent forces as director forces.
Any parabolic group with 61^ as director line is semi-latent. Such a
group is either latent, or has only one latent system (the director force).
For, let a and b be any two (latent) points on Cie^, and let x and y be
334 MATRICES AND FORCES. [CHAP. IV.
any other two points. Then <f>a = yia, ^ = 716, <^ = 7ifl? + a/, ^y = 7iy + y';
where x' and y' are points on ^i^s. Hence, if ^ be the system ax -{-by,
since ax' = ^ = ^^
Thus 5 and €16^ define a semi-latent parabolic group. If /8> be latent
(that is, if \ be zero), the group is latent.
The semi-latent triple groups (not entirely single forces) are of two
kinds, which will be called, type I. and type II.
A semi-latent group of type I. is defined by any three non-intersecting
latent forces, namely by x<l>x, y<l>y, z<f>z. Any such group is latent.
The groups of type 11. are the groups reciprocal to those of type I. Thus
^i^s, which is a common null line of every group of type I., is a common
director line of every group of type II. Also, since every other semi-latent
line intersects ei^s, this force is the only latent director force of any group
of type II.
Again it has been proved that every semi-latent line is a null line of the
system 816164 -{-S^e*^. Hence this system is reciprocal to every system of
every semi-latent group of type I. Accordingly the semi-latent parabolic
subgroup defined by Ci^j, B16164 + S^e^^ is a subgroup of every reciprocal
semi-latent group of type II. The force 6i6t is the only latent system of
this common subgroup, since the system B16164 + h^s^^ is not latent. Thus a
semi-latent group of type II. has only one latent system, namely e^e^. For,
by § 175 (9), a director force {e^e^ of a triple group can only be a null line
of one dual subgroup of systems; and 616^ is a null line of every latent
system.
Semi-latent groups of type II. are defined by 616^, 816164 -{-S^^, and any
system 8. For the system S can be written
Hence ^8 = y^^S + («rj,8i7i + ^u^^i + ^2i^8») ^i^s H- '0^a47i (^16164 + B^^i).
Thus <l>8 belongs to the triple group defined by S, 616^ and S16164 + S^^.
The other semi-latent parabolic groups cannot belong to any semi-latent
triple group. For the reciprocal to such a group must be another semi-
latent triple group. But groups of type I. and II. are respectively reciprocal
in pairs : so this reciprocal semi-latent group must contain another semi-
latent parabolic subgroup. Thus two reciprocal triple groups would each
contain the director line 616^ in common. This is impossible for triple
groups, not of a special kind.
(6) Let the roots of the matrix 0 be equal in pairs ; so that 71 =» 7,, and
73 = 74. Then [cf. § 190 (5)] there are three cases to be considered, as far as
the distribution of latent and semi-latent points and regions is concerned.
198] ENUMERATION OF LATENT SYSTEMS AND SEMI-LATENT GROUPS. S85
But each of these three principal cases gives rise to another case, in which
the additional relation 71 = — 7t is fulfilled. Thus there are six cases in all.
Cdse /. [Cf. § 190 (5), Case I.] The latent regions are the lines e^e^
and 68^4. The latent systems are the forces 6162 > ^a> ^^^ &i^7 system of
the quadruple group which has ^6, and e^^ as common null lines.
The semi-latent dual groups are defined by any two of these latent
systems; since by § 192 (14) no semi-latent parabolic group exists with
only one latent sjrstem.
The semi-latent triple groups are defined by any three of these latent
systems.
There are only two types of semi-latent triple groups, of a general kind :
let them be called type I. and type 11.
Type I. consists of groups defined by three systems of the form, oft, a'h\
a"V\ where a, a\ a" lie on 61^, and 6, h\ h" lie on e^^.
Type II. consists of groups defined by three systems of the form eyfi^^ 6^4,
"Kab H- fia^V. E^h group of type II. is reciprocal to a corresponding group
in type I.
Groups defined by two forces of the form ab and a'6', together with either
61^2 or ^64, are not of the general kind ; since in them the director force 616^
(or e^e^) intersects the director forces ab and a'b\
Case II. [Cf. § 190 (5), Case II.] The latent regions are the line eiC^
and the point ^. The latent systems are, the forces eie^, e^^y and any
force of the type ez (\«i + fie^» There can be no other latent systems ; for
the coplanar lines of the type e^i^ei^- fjis^ and 61^2 are not all concurrent
and therefore cannot all be null lines. Hence by using § 192 (3) the pro-
position is easily proved.
There is a semi-latent dual group defined by ^e^ and 6^4.
There are semi-latent parabolic groups with any force of the type
61 (M + /Ae«) as director line [cf § 192 (14)].
Now by § 192 (11) the point X^H-/a€8 is the null point with respect
to the group of the plane eie^ ; and the point e^ is the null point of the
plane (Xei + M^) ^9^4 ; since both e^e^ and e^^ must be common null lines
of the group. Hence any other system of such a group must be of the form
/8f = e, (Xei -f /tea) + f^a^s + 1; (X«i + m^) C4.
And by § 190 (5), equation (K), it follows that
Hence any parabolic dual group of the type defined by ^(X«i + a^)
and f eg&s + 17 (X^i -h M^) ^4 ^ semi-latent; and all such semi-latent groups
contain only one latent system.
There can be no semi-latent triple groups of the general type [cf § 192
(17)].
886 MATRICES AND FORCES. [CHAP. IV.
Case III. [Cf. § 190 (5), Case III.] The only two latent points are
ei and e^. The latent systems are ^i^s, e^^y eie^, and any system of the type
^^1 + /* (7i^8^2«8 — 7381^1^4)- For [c£ § 192 (3) and (4)] the point ^ is the
null point of the plane eie^, and the point e^ is the null point of the plane
6^461, Hence any possible latent system S (not a single force) must be of
the form
Now [cf § 190 (5), equation (L)]
<^S = 7i78iS H- (YgSifta + 718,^4) ^^j (1).
Thus, if S be latent,
fhs-yiS^, fti4 = -7«^-
The semi-latent dual groups (not consisting entirely of single forces) are,
the group defined by BiB^ and 6^64; any parabolic group [cf. equation (1)] of the
type defined by the director force €16^ and a system of the form /j^^^e^ + /ii4^&4;
any parabolic group of the type defined by BiB^ and Xb^ + fi (yiS^e^^ — 78816164) ;
any parabolic group of the type defined by b^4 and Xb^ + fjL {yiB^e^fi^ — 73816164).
It can easily be proved that no other dual groups are semi-latent, as
follows. The only possibility lies in the existence of a parabolic group with a
single latent system. But, from § 192 (11) and (14), 61^ is the only possible
director force of such a group ; and the group must have the form stated
above. Further, by equation (1) above, the group stated is actually semi-
latent.
In searching for triple groups of the general type, it is useful to notice
that such a group cannot be defined by a director force D and two systems
8 and ST, such that (D8) = 0 = {DSy For by § 175 (9) in a triple group of
the general type only one parabolic group with D as director force can exist.
The semi-latent triple groups with three latent systems are all groups of
the type defined by 6162, 6364, X6861 + fi (yiB^e^s ^ yJ^i^i^*)- Call such groups the
semi-latent groups of type I.
No two groups of type I. can be reciprocal to each other, since all such
groups have one pair of director forces in common, namely 616, and ^4.
Thus there must be another type of semi-latent groups. Call them the
groups of type II. The only semi-latent subgroups, which a group of tjrpe II.
can contain, are parabolic subgroups of the type defined by 6163, /Aas6a^+/^i46i64.
Now the condition that the system ^£386263 + Ati46i64 may be reciprocal to
the system X6s6i -1-/11(718,626, — 7,816164) is
7i^«/^4 - 78^1/% = 0.
Hence all groups of type II. contain the parabolic subgroup defined by
616,, 718,686, + 7,816164. There is no other latent system which, in conjunction
with this subgroup, will define a triple group of the general type. Hence
all groups of type II. contain only one latent system, namely 616,.
193] ENUMERATION OF LATENT SYSTEMS AND SEMI-LATENT GROUPS. 337
Any system 8, which has ^e, and e^^ as null lines, defines in conjunction
with the systems ^i^, yA^fy + 7s^6i^4, a semi-latent triple group of type TI.
For we may write
Then if>S = 7i7,Sf + (istsAt* + ^'i/yi^s + «r94^§8) «i^ + ^'ai (yi^Afiz + 7^^164).
Hence ^/S belongs to the required group. There can be no other semi-
latent triple groups. For any other semi-latent triple groups must contain
a parabolic subgroup of the type €169, fJi'^^B^ + fhA'h^A' But two such triple
groups (of the general type) cannot be reciprocal to each other, since they
both contain a common director force ^e^.
Case IV. (Subcase of Case I.) The latent and semi-latent points and
regions are the same as in Case I. : but the additional relation 71 = — 7s is
satisfied. Thus 71* = 7,*
The latent systems are the forces e^e^^ e^4, any system of the type
Xeie2+/i^4, and any system of the quadruple group which has ^e^ and
e^4 as common null lines. Any latent system 8 is either such that
^ = y*8 = 7,»fif ; or such that <f>8 = 7i7,S = - 7i«S « - 7/S.
No semi-latent parabolic group exists with only one latent system [cf.
§ 192 (14)] Hence all semi-latent groups have their fiill number of latent
systema
The semi-latent dual groups are defined by any two of these latent
systems. The semi-latent triple groups are defined by any three of these
latent systems.
There are three tj^pes of semi-latent triple groups. Type I. and type II.
are the same as in Case I. ; and their groups are reciprocal in pairs. Type
III. consists of groups defined by three latent systems of the form X^e, + fie^^,
aby a!h\ The groups of this type are reciprocal in pairs ; since the group
defined by Xaie9 + A^s^4> 06, o!h\ is reciprocal to the group defined by
Cobb V. (Subcase of Case II.) The latent and semi-latent points and
regions are the same as in Case II. : but the additional relation 7^ = — 7, is
satisfied. The latent systems are, the forces de^, ^4, any system of the type
There is a latent dual group defined by eie^ and ^4 ; semi-latent parabolic
groups defined by latent systems of the type X^ei + /i6|04 &nd Bt{\'ei + fie^);
and semi-latent parabolic groups, with only one latent system, defined by
systems of the type ^(X^ + /a€^) and ^6^ + 17 (X^ 4-/^)64 [cC Case II.].
There can be no semi-latent triple groups of the general tjrpe
[c£ § 192 (11)1
W. 22
338 MATRICES AND FORCES. [CHAP. IV.
Case VI. (Subcase of Case IIL) The latent and semi-latent points and
regions are the same as in Case III.: but the additional relation ^1 = — 7» is
satisfied
The latent systems are the same as in Case III. with the additional set of
latent systems of the form X^^s + fjue^^.
The semi-latent dual groups (not consisting entirely of single forces) are,
the (latent) group defined by e^e^ and ^4 ; any parabolic group of the type
defined by 61^ and {jl^^^ + /^4^«4 5 any parabolic group of the type defined by
ei^a and Xe^i + /A (yiS^e^i — yiK^ie^ ; any parabolic group of the type defined
by ^4 and \e^ H- fi {^i^^R^ — yJ^iBie^) ; and any group of the type defined by
Xei^ + fie^4 and \^e^ + /a' (7183^3 — y^^'^i'^d-
The semi-latent triple groups are simply those of type I. and type IT. in
Case III. For the only possibility of additional semi-latent triple groups (of
a general kind) beyond those of Case III. lies in the semi-latent groups of
the type defined by X^Cj + ^«^4, \'e^ + /*' (ji^^s^ — 78^16164), ^63. But these
triple groups are not of the general kind, since the director force ^63 is the
director force of two parabolic subgroups belonging to such groups.
194 Transformation of a Quadric into itself. (1) When a triple
group is semi-latent, the matrix must transform the director generators of
the associated quadric [cf. § 175 (4) and (5)] into director generators of the
same quadric. Thus each point on the associated quadric is transformed into
a point on the same quadric ; and the quadric may be said to be transformed
into itself by a direct transformation, the associated triple groups being semi-
latent.
(2) There is another way in which a matrix may transform a quiGulric
into itself, so that the associated triple groups are not semi-latent. For the
generators of one system may be transformed into generators of the other
system, and vice versa. Let this be called the skew transformation of a
quadric into itself, and let the first method be called the direct trans-
formation.
(3) If a matrix transforms a quadric into itself, by either direct or skew
transformation, then every semi-latent line either has two distinct latent
points on it, or touches the quadric.
For let any semi-latent line cut the quadric in the two distinct points p
and q. Then ^ and ^ are also on the quadric and on the semi-latent line.
Hence either <l>p=p, ^ = 9, in which case p and q are two distinct latent
points on the line, or if>p^q, <l>q = p. In the second case (f>^ = (f>q=p, and
ipl^^q.
Now if e be the sole latent point on the line, and 7 be the repeated
latent root corresponding to e, then ^ = yp + Xe, and <f>^ = 7^ H- i'X/ye = p.
Hence X = 0. Thus p is a latent point. Similarly 5 is a latent point. Thus
l^ 194, 195J TRANSFORMATION OF A QUADRIC INTO ITSELF.
339
the line is a latent region corresponding to the repeated root, and e is not
the sole latent point on it.
But if the semi-latent line touches the quadric, this reasoning does not
apply. The point of contact must be a latent point; and, as far as has been
shown, it may be the only latent point on the line.
(4) If the semi-latent line does not touch the quadric, and if if^ = q,
<l>q=p, assume that ei and e^ are the two distinct latent points on the line,
and that 71 and 7, are the corresponding latent roots.
Also let p = Xa^ + \^. Then if>*p = 7i»XiCi + yi*\^ = Xi^i + \^. Hence
7i" = 72** But if 7i = 7a, then <f>p=p, <f>q = qt which is contrary to the assump-
tion. Hence in the present case 7^ = — 7,; and ^ = 71 (\ei — X^^) = q. Thus
(pq, €16^) form a harmonic range. Thus ei and e, are conjugate points with
respect to the quadric.
(5) The polar reciprocal of a latent or semi-latent line is itself latent or
semi-latent. For since the quadric is unaltered by transformation and the
original line retains the same position, its polar reciprocal must also retain
the same position.
Also if a point be latent, its polar plane must be semi-latent. Hence
if the latent point be not on the quadric, at least one other latent point
must exist on the semi-latent polar plane. Then the line joining these
latent points is semi-latent; and its reciprocally polar line is also semi-
latent.
196. Direct Transformation of Quadrics. (1) It follows from the
enumeration of § 193, that the only cases in which semi-latent triple groups
of the general type exist are those cases stated in § 193 (2), Cases I. and II. :
in § 198 (3) Case III. : in § 198 (5), Case V. : in § 193 (6), Cases I. and III.
and IV. and VI. In all these cases the relation, 77' = y'y", holds between
the four latent roots of the matrix. In the two cases of § 193 (2) the four
latent roots are distinct ; and there are only four latent points, which form
a tetrahedron. In Case III. of § 193 (3) two latent roots 71 and y^ ai*o equal,
and 7i' = yff4 : also a latent line exists corresponding to the double root : this is
really a subcase of Case I. in § 193 (2). In Case V. of § 193 (5) all the latent
roots are equal, there is a latent line, and an infinite number of semi-latent
lines intersecting the latent line. This is really a subcase of Case III. of
§ 193 (6), and will be discussed after that case.
In Cases I. and III. of § 193 (6), the latent roots are equal in pairs, namely
7i='7s> 78 = 74; ^>^^ either (Case I.) two latent lines exist corresponding
respectively to the two distinct roots ; or (Case IH.) only two latent points
exist, one corresponding to each root. Case I. is a subcase of § 193 (2).
Case IV. of § 198 (6) is Case I. with the additional relation 71 = -7,.
It is partly merely a subcase of (6) Case I.: but it also transforms other
22—2
340 MATRICES AND FORCES. [CHAP. IV.
quadrics according to the type of (2) Case I.: thus it is partly a subcase
of (2) Case II.
Case VI. of § 193 (6) is a subcase of Case III., and in no way differs from
it in its properties with regard to the direct transformation of quadrics.
(2) The general type of direct transformation of quadrics is given by
§ 193 (2), Case I. Then the associated quadric of any group of the type
defined by 6164, e^^ o^ez^-fie^^ is transformed into itself by direct transfor-
mation. The reciprocal group, associated with the same quadric, is eie^y
6^A9 a^^~i8^9^4» ^^d this group is also semi-latent.
The semi-latent lines ^^4, e^ are generators of one system, and the
semi-latent lines ^^, ^4 are generators of the other system. Hence the
quadric has four of its generators semi-latent, two of one system and two of
the other.
It follows that the semi-latent lines ^e, and e^^ are reciprocally polar
to each other, so that the polar plane of any point on one contains the other.
All quadrics containing these four geuerators are transformed into them-
selves. For they are defined by either group of a pair of reciprocal triple
groups of the t3rpes mentioned above.
(3) In Case II. of § 193 (2), the only difference from the general case
is that an additional set of quadrics are transformed into themselves ; namely
the associated quadrics of groups of the tjrpe defined by e^et, ^4, ob^e^ + fie^t-
The recipix)cal group associated with the same quadric is 61^, 6^4, OLeie^—^e^)
and this group is also semi-latent.
(4) In Case III. of § 193 (3) the only difference from the general case is
that any two points Oi and a, on the line eiC^ can be substituted for ^ and e,.
Then Oi^s, 0^4, 01^4, a^ are generators of one of the quadrics; and ^^ and
e^4 are reciprocal lines.
(5) Case I. of § 193 (6) is really only a subcase of that described in the
subsection (3). Let Oi and a, be any two latent points on the line 61^;
and let a, and a^ be any two latent points on the line e^^. Then the
explanation of the previous subsection applies, substituting any tetrahedron
such as OiO^a^^ for the tetrahedron e^e^^^ in the previous subsection. Thus
all quadrics which have the two latent lines e^e^ and ^4 as generating lines
are transformed into themselves by direct transformation. Also all such
quadrics have two generators of each system latent, or semi-latent.
(6) Case IV. of § 193 (6) transforms into themselves the quadrics
mentioned above both in (3) and in (5): that is, quadrics with the latent
lines 6169 and e^^ as generating lines, and quadrics with 61^2 and ^4 as
reciprocal lines. The transformation may be represented as follow& Let x
be any point; draw through m the line 4^ intersecting exe% and ^4 in p
and q. Then x^ <^, p, j form a harmonic range.
195] DIRECT TRANSFORMATION OF QUADRlCa 341
Let flf = ^6, + e^4, iSf' a= CiCa — ^4- Then [cf. § 179 (4)J we may put
(7) In Cases III. and VI. of § 193 (6) the semi-latent triple groups
belong to two t3rpes ; and are reciprocal in pairs, one from each type. A
typical specimen of type I. is defined by ^^a, e^^, Xe^ + fi (yiS^e^ - ^it^e^e^ :
a typical specimen of type II. is defined by
^169, 7i^s^sA + 7A^«4> ^2Afi» + ©"si^i + «^i4«i«4 + «ra4«a^4.
Hence all quadrics, which are transformed into themselves, have two
semi-latent generators of one system, namely ^^ and e^^; and one semi-
latent generator of the other system, namely Cie^, All the quadrics, so
transformed, touch each other along the generator ^i^ ; since [cf. § 187 (8)]
the parabolic subgroup, defined by Cie^ and yiB^B^ + jt^ieie^, is common to
them all
The systems of the type \e^ + fi (yiS^e^ — yzBiBie^) have only three null
lines which are generators of the quadric associated with the triple group,
defined by any one of them together with €162 and e^^ [cf. § 175 (13)]. Thus
consider the quadric defined by ^6^, ^4, and e^ + v (yAe^ — Jz^'^^a)' Let
it be called the quadric A ; also for brevity write S=p (71^16^ — yJ^iBie^).
Then we have to prove that the system /S' = X^i + /a (718,626^ — 7,8161^4) has
only three null lines which are generating lines of the quadric A ; unless
\v=ifA. For take the quadric A as the self-supplementary quadric, then
[cf. § 175 (8)] we may assume that
Hence 1 8 = 2e^ + 8. Also it is easy to see that (eieJ3) = 0.
Now v8' = Xve^^ + fi8. Hence i/» (8' 8') = /*• (88).
Again v |iSf' = \i/ \e^ + /* [S = (2/* — \v) e^ + fi8.
nence v'(S'\8')^fi' (88) ^p" {8^8).
Thus (8' \8') = (iST/ST). But [cf. § 175 (13)], this is the condition that 8'
may have only three null lines which are generators of the quadric A,
assuming that \8'^8\
(8) Case V. of § 193 (5) is a subcase of the case discussed in the
previous subsection. The semi-latent triple groups belong to two types,
such that the groups are reciprocal in pairs, one from each type. * A
typical group of type L is defined by three latent forces x(l>x, yffsy, z<f>z: a
typical group of type II. is defined by 61^, 816164 -h Sgeg^, 8\ where 8 is any
system. Thus any quadric transformed into itself has the latent line ejfit
as a generator, and has no other latent or semi-latent generator of the same
system; also all the generators of the opposite system are semi-latent
[cf. § 192 (17)]. All quadrics which are transformed into themselves touch
along the generator 616^, since [c£ § 187 (8)] they have a common parabolic
subgroup with eie^ as director force.
342 MATRICES AND FORCES. [CHAP. IV.
(9) If the four latent roots 71, 72, 78, 74, satisfying the relation 717, = 7^4,
be assigned, then a matrix can in general be constructed, which will transform
the quadric into itself by direct transformation, and at the same time make
an assigned system latent.
For let 8 be the assigned system. Then [cf. § 174 (9)] 8 has in general
one pair of conjugate lines which are polar reciprocal with respect to the
given quadric. Let ei^ and e^e^ be these lines, cutting the quadric in the
points ^1, 62, e,, e^. Consider the matrix for which ei, e^, Cj, e^ are the latent
points corresponding to the latent roots 71, 7a, 7s, 74. The given system is
obviously latent, since it is of the form aei6s + )8es«4. Also all quadrics con-
taining the four generators ^^4, e^y 61^3, 63^4 are transformed into themselves,
and among them the given quadric.
(10) But if the system 8 has only three null lines, which are generators
of the quadric to be transformed into itself; then the matrix must be of the
type of § 193 (6) Case III., or must belong to one of the subcases. Then [c£
subsection (7)] with the notation of § 193 (6) Case III., let ^i, 62, ^> ^4 be so
chosen that the three generators, which are null lines are, e^e^, 616^, e^^- The
system 8 can be written e^ + ae^ + ^61^4, where a and )8 are known, since 8
is known. Then 74, 7,, 81, S, must be so chosen that
a^_7iS,
Also by an easy extension of subsection (7) [cf. § 175 (13) and § 187 (8)]
the quadric is defined by three systems of the form
Hence the quadric is transformed into itself, at the same time as iSf is latent,
by the operation of the matrix.
(11) Thus from (9) and (10), it is always possible to find a matrix which
transforms directly a given quadric into itself, and keeps a given system
of forces latent. And the matrix is not completely determined by these
conditions.
196. Skew Transformation of Quadrics. (1) When a quadric
is transformed into itself by a skew transformation, no generator can be
semi-latent.
(2) If ^ be a matrix which transforms a certain quadric into itself by a
skew transformation, then the matrix <f>^ transforms the same quadric into
itself by a direct transformation. It is usefol to notice that the latent points
of ^ are also latent points of <f>\ though the converse is not necessarily true.
(3) Let the matrix ^ have four distinct latent roots. Let 71, 73, 7,, 74
be the distinct latent roots, and 61, 62, ^, e^ the corresponding latent points.
Then the latent roots of <f>^ are 71*, 72', 78^ 74*, and 61, 6^, e,, C4 are latent
points. Now either 71", 72", 78*, 74' are distinct; or, two are equal, y^vszyf^
so that 7i = — 7a ; or, they are equal in pairs, 71" = 7,', 7,' = 74*. •
196] SKEW TRANSFORMATION OF QUADRICS. 343
Hence <^« is either of the type of § 193 (2), or of § 193 (3), Cases I.
or III., or of § 193 (6), Case I. Case II. of § 193 (3) cannot occur because
[cf. subsection (2)] the line ^^2 has two null points, ei and 6^, on it. Similarly
the other cases of § 193 (6) cannot occur : § 193 (6) Case IV. is inconsistent
with the roots being distinct.
But <!>* transforms the quadric into itself by direct transformation.
Hence § 193 (3), Case I. is impossible ; and § 193 (2) and § 193 (6), Case I.
both make semi-latent lines of ^ to be generating lines of the quadric, which
is impossible by subsection (1) above.
If the additional relation, 71^ =3 7s'7A hold, then <f>^ is the type of matrix
described in § 193 (3) Case III. The latent roots of ^ are connected either
by the relations 71 = — 7, = V'7t74> or by 71 = — 7, = V— 7»74» With the nota-
tion of § 195 (4), the points Oi and a, are on a quadric transformed directly
into itself by ^^; and Oje,, 0164, a^, 0^4 are generator of this quadria
Hence, if (f> transforms this quadric into itself by a skew transformation,
Oi and Oa cannot be latent [cf. subsection (1)].
Hence, since they lie on a semi-latent line, ^Oi = a^, ifxt^ = Oi. Hence,
by § 194 (4), (oiOs, ^eg) forms a harmonic range. Also (fxii == 71a,, ^o, = 7101.
Now for the quadric defined by the group OiO, + Xe^^, Oi^, 0^4, to be trans-
formed by ^ by a skew transformation, this group must be transformed by <f>
into the reciprocal group OiOs — X^4, a^, Onfii.
Now if> (OjOa + \«,64) =: - 7i"aiaa -h y//M^4, (fxh^t = 7i7»^s^»» <A«a^4 = 7i74«i«4-
Thus it is necessary that 71* = 7^4.
Hence a matrix with four distinct latent roots, related so that
7i = -7» = V7874,
transforms into themselves by a skew transformation quadrics, passing
through ^ and 64, with CiC^ and e^4 as polar reciprocal-lines, and with ei and
€2 as polar reciprocal points.
(4) Let the matrix ^ have three distinct roots. Assume 61, ^, $$, 64
to be such that
^=7i^> ^ = 7i«, + 8iei, ^ = 7«et, ^4 = 7A;
where Si may, or may not, be zero.
Hence ^^ = 71^, ^«, « 7i*ea + 2Si7iei, <!>% == 7,*^, ^64 = 74*^4.
Now four cases arise.
Case A. Let 71", 7,*, 74* be distinct Then the matrix ^ is of the type
described in § 193 (3). Hence it cannot transform a quadric by direct
transformation into itself; except in Case III. But in § 193 (3) Case IIL
the lines of the type Oi^, ajC4, ai«4, cMi [cf. § 196 (4)] are generators of the
transformed quadrics. But these lines are semi-latent lines of ^ as well as of
^': and hence [cf. subsection (1)] this case must be rejected.
344 MATRICES AND FORCES. [CHAP. IV.
Case B. Let yj* = */,'; so that 71 = — y,. Then the matrix ^' is of one
of the types (Cases I. and II.) described in § 193 (4) ; either it is Case I. if Bi
vanish ; or it is Case II. if Si do not vanish. In either case ^' cannot trans-
form a quadric into itself by direct transformation.
Case G. Let 78* = 74* 5 and ^ be not zero. Then <f>* is of the type
described in § 193 (6), Cases IL and V. The other cases of § 193 (6)
cannot occur, since the three latent roots 71, 79, 74 are by hypothesis distinct ;
and the points e^ and 64 are both latent points of <f>\ Hence <f>* cannot trans-
form a quadric into itself by direct transformation.
Case D, Let 7,' = 74' ; and Si = 0. Then «^" is of the tjrpe described
in § 193 (6), Cases I. and IV. If 0« belongs to the type of § 193 (6)
Case I., then by § 195 (5), ^' transforms into themselves all quadrics with
61^2 and 6364 as generating lines. But these are semi-latent lines of ^.
Hence by subsection (1), this case is impossible.
But if ^« belong to the type of § 193 (6) Case IV., so that
then ^ transforms quadrics directly for which CiC^ and e^^ are polar reciprocal
lines. Thus since 73' = 74* = — 7i7ai we have a subcase of the transformation
considered in subsection (3). But it is the alternative case for which ^ does
not effect a skew transformation.
(5) Assume that ^ has two distinct roots, one root 71 occurring triply.
Let ^, 6^, Bzi 64 be assumed so that
0Ci = 7i6,, 0e2 = 7A + ^i^* ^ = 7i«« + ^a^> <^«4 = 74«4;
where Si and S, may, or may not, vanish.
Then
<^^ = 7i»^, ^e, = 7i*^ + 2S,7iCi, ^«, = 71"^ + 2Si7ieg + SiS^ei, 0»C4 = y4%.
Let the point e^' =^ 271^ — S^. Then e^ 62, e^ are such that
0*«i = 7i"^> ^'^2 = 7i'^+2Si7iei, ^V = 7i'^' + 4%"«a-
Case A. Let 74' be not equal to 71". Then ^ must be one of the three
types described in § 193 (4). But in no one of the three cases of that article
does the matrix transform a quadric into itself by direct transformation.
Case S. Let 74 = — 71. Then ^ must be one of the types described in
§ 193 (5). The matrix ^« is of the type of Case IIL of § 193 (6), if Si and S, do
not vanish : it is of the t3rpe of Case II., if Si vanishes : it is of the type
of Case I., if Si and S, both vanish. But in Cases IL and III. no quadric
is transformed into itself by direct transformation. In Case I. the matrix <f>^
is merely the numerical multiplier 71". Hence every quadric is transformed
into itself, since no point changes its position.
Then the matrix 0 has two latent roots 71 and -71. There is a latent
196] SKEW TRANSFORMATION OF QUADRICS. 345
plane eie^ corresponding to 71; and a latent point ^4, not on 61^1^,, corre-
sponding to — 7i. These are the only latent regions. Hence, by § 194 (6),
for all quadrics which are transformed into themselves by ^, e^ and BiB^ must
be pole and polar. Also e^ cannot lie on such a quadric, since it does not lie
on its polar plane.
Now, if /> be any latent point on the plane CiC^, and «? = X64 + /Ltp, then
Hence [e^p, x^] forms a harmonic range. Thus if a; be a point on a
quadric for which e^ and BiB^ are pole and polar, ^ is also on the same
quadric. Also the transformation is skew, since by § 192 (16) it cannot be
direct. This is a subcase of the skew transformation of subsection (3), since
7i=-74 = V7a78-
(6) Assume that ^ has only one root. Let ^, e^, e^, e^ be assumed
so that
Then ^^ = 7i'^, ^«a==7i*^2 + 27i8iei,
<l>% = 7i'^ + 27aSag, + SiS^i, <!>% = 71*^4 + ^i^A + ^^i-
Let Ci' = 271^ - Sa^j, 64 = ^1% - 27i8s^ + 82^362.
Then if>%' = 71V + 47i»8^, <^; = 71V + ^i^^^
Hence 0' is a matrix of one of the tjrpes described in § 193 (6). The
case, when Si = S, = ^=0, need not be considered: for then ^ is a mere
numerical multiplier. Thus [cf. § 195 (1)] the only case of this type in
which <f>* transforms a quadric into itself by direct transformation is that of
§ 193, Case Y. Then Si and ^ do not vanish, and £2=0. Li this case e/^e^;
also [c£ § 195 (8)] the latent line eiC^ is a generator of all quadrics trans-
formed by ^'. But 61^ is also a latent line of ^. Hence if> cannot transform
these quadrics into themselves by a skew transformation.
(7) Assume that the latent roots of 0 are equal in pairs, so that 71 = 7,,
and 7s = 74. Let Ci, 62, e^, e^he such that
^ = 7i^, ^ = 7iCa+Siei, ^ = 78&t, ^«4 = 7A + 8,^.
Then ^ei = 7i«ei, <f>%=^yi%'k'2yAei, ^'^=7,»e., ^I'e^^yt^e^ + Zy^^.
Hence ^' belongs to the type described in § 193 (6). Of the six cases
of this tjrpe only Cases L and IIL and IV. and VL yield quadrics which are
transformed into themselves by ^' with a direct transformation. In Cases
I. and IV., Si = 0 = S,, and [cf. § 195 (5) and (6)] either Cie^ and ^4 are
generating lines of such quadrics, or they are reciprocally polar lines to them.
If they are reciprocally polar lines, the four semi-latent lines, joining the two
pairs of points in which eie^ and ^4 meet any such quadric, are generating
lines of the quadric. But the latent lines Cie^ and ^4, and the semi-latent
lines joining any point on Cie^ to any point on ^4, are latent and semi-latent
346 MATRICES AND FORCES. [CHAP. IV.
lines of ^ as well as of ^^ Hence if> does not transform any of these quadrics
into themselves by a skew transformation.
In Cases III. and YI. of § 193 (6) neither Si nor S, vanishes. All
quadrics transformed into themselves by ^" have [cf. § 195 (7)] the three
lines 6162, 6162, e^A as generators, which are semi-latent with respect to ^ as
well as with respect to ^^ Thus if> does not transform these quadrics into
themselves by a skew transformation.
(8) Thus there is only one case of skew transformation, namely the
case (including its subcase) when
7i = - 7a = V7i74 ;
and the subcase arises when
7i = 72 = 7« = -74.
In the general case the lines eiC^ and ^4 are polar reciprocal with respect
to any quadric so transformed, the points Ci and €2 are polar reciprocal, and
the points e^ and e^ are on the quadric (except in the subcase, when y^ = 74
and the line e^^ is a latent region).
In the subcase the point e^ and the latent plane CiC^ are pole and polar
with respect to all quadrics so transformed.
NoTB. Homersham Cox, On the Applicatwn of Quatemhns and Orassmarm^s Algebra
to different kinds of Uniform Space^ Trans, of Camb. PhU. Soc., 1882, points out the
connection between a positional manifold and Descriptive Geometry of any dimensions
[cf. Book III.], and applies it to Hamilton's theory of nets. Also he points out the special
applicability of Outer Multiplication to Descriptive Geometry [cf. Chapter IV., Book IV.] ;
this had already been practically demonstrated by Grassmann in his papers in OrelU^s
Journal on Gubics. Further [in correction of note, p. 278] he applies the calculus in the
manner of this book to deduce some elementary propositions concerning Linear Complexes ;
he finds the condition for reciprocal systems [cf. § 116 (1)], for null lines [cL § 163 {!)},
the director equations of dual and triple groups [cf. § 172 (1) and § 175 (1)], and the
condition for a parabolic group [cf. 172 (9)]. He also finds a defining equation of Intensity
[cf. note, p. 168], which depends on the distance between points. The bulk of this very
suggestive paper is concerned with the Theory of Metrics.
BOOK VI.
THEORY OF METRICS.
•I
CHAPTER L
Theory of Distance.
197. Axioms of Distance. (1) In a positional manifold, to which
no additional properties have been assigned by definition, no relation
between any two points can be stated without reference to other points
on the manifold. Thus consider a straight line which is a one^imensional
positional manifold. If ^ and e^ represent the reference elements at unit
intensity, any point p can be written fiei + fa^- But fi is not the ex-
pression of a quantitive relation between p and 61. For fi depends on ^9,
^^d fi/fs represents a relation of p to the terms ei and eg. But even this
does not properly represent a relation of the element represented by j> to
those represented by ^ and eg. For no determinate principles have been
assigned by which the terms Ci and eg should be considered to represent
their corresponding elements at unit intensities. Thus the arbitrary
assumption as to the intensities is included, when fi/f, is considered as
representing a quantitive relation o{p to 61 and e^.
The only relations between points, which are independent of the
intensities, are the anharmonic ratio between four points [cf. § 69 (1)], and
functions of this anharmonic ratio.
(2) A spatial manifold will be defined to be a positional manifold,
in which a quantitive relation between any two points is defined to exist.
This quantitive relation will be called the distance, and the following
axioms will be assumed to hold of it.
(3) Axiom I. Any two points in a spatial manifold define a single
determinate quantity called their distance, which, when real, may be
conceived as measuring the separation or distinction between the points.
When the distance vanishes, the points are identical.
Axiom II. If p,q,rhe three points on a straight line, and q lie between
p and r [cf. § 90 (3)], then the sum of the distances between p and q and
between q and r is equal to the distance between p and r.
350 THEORY OF DISTANCE. [CHAP. 1.
Axiom III. If a,b,che any three points in a spatial manifold, and the
distances ah and be be finite, then the distance ac is finite. Also if the
distance ah be finite and the distance be be infinite, then the distance ac is
infinite. Also if the distances ah and be be real, then the distance ac is also
real.
(4) Let pq be any straight line through p ; and assume some rule to
exist, by which one of the two intercepts between p and any point q on the
line can be considered as the intercept [cf. § 90] such that points on it lie
between p and q ; then points on this line on the same side o{ p bs q are
points which either lie between p and q or are such that q lies between
them and p. It follows from axiom II. that all points between p and q are
at a less distance from p than the distance pq, and that all points on the
same side as q, but beyond q, are at a greater distance firom p than is q.
Also it is evident that there cannot be another point on pq on the same
side o{ p BS q and at the same distance as q. For if g^ be such a point,
then by axiom II. the distance qq' must be zero ; and hence by axiom I. the
points q and ^ coincide.
(5) Hence the relation of a point g to a point j> in a spatial manifold
is completely determined by (a) the straight line through p on which q lies,
(fi) the determination of the side of p on which q lies, (7) the distance of q
from p. Thus any quantitive relation between points on a straight line
must be expressible in terms of their distance.
198. CoNGEUBNT RANGES OF POINTS. (1) Two ranges p, q, r, 8, ...
and p\ q\ r^, s', ... of the same number of points in a spatial manifold are
called congruent when the following conditions hold. Let the points p, q, r,
8, ... and the points p\ q', /, 8\ ... be mentioned in order; also let the
distance between p and q be equal to that between p' and gf, and the
distance between q and r equal to that between ^ and /, and so on.
(2) It follows from this definition of congruent ranges and from axiom
II. of distance, that the distance between any two points on one range is
equal to that between the corresponding points on the other range.
(3) Also [cf. § 197 (5)] any quantitive relation between points in the
first range, which can be expressed without reference to other points of the
spatial manifold, is equal to the corresponding relation between the corre-
sponding points of the second range. Such a relation in a positional
manifold is the anharmonic ratio of a range of four points. Hence con-
gruent ranges must be homographic [cf. § 70].
(4) Also conversely, if on two homographic ranges the distances between
three points of one range are respectively equal to the corresponding
distances between the three corresponding points on the other range, then
the ranges are congruent. For let pqr and p'^r' be the two groups of three
198, 199] CONORUENT RANGES OF POINTS. 851
points on the two ranges, and let 8 and s' be any other two corresponding
points on the ranges respectively. Then the anharmonic ratio of (pqrs)
equals that of (pV^*')- ^^* ^^ ^^^ range (p'^r^^^ be constructed congruent
to the range (pqrs), then by the previous part of the proposition the
anharmonic ratio of (pqrs) is equal to that of (p'q'r^tfy Hence the an-
harmonic ratio of (p'^r^s") is equal to that of (pY^'O* Hence s' and s"
coincide.
The proposition may be stated thus, if three points of one range are
congruent to the three corresponding points of a homographic range, then
the ranges are congruent.
199. Cayley's Theory of Distance. (1) Cayley has invented in his
'Sixth Memoir on Quantics'* a generalized expression for the distance
between two points of a positional manifold. This work was extended and
simplified by Klein "f, who pointed out its connection with Non-Euclidean
geometry.
(2) Consider in the first place a one-dimensional region. Let a^ and a,
be two arbitrarily assumed points on it. Then any three points Xi,x^, and x^
of the region can be written X^Oi + ftiOa, X/ij + fi^, \^ + fi^.
The anharmonic ratios pn, pu$ Rn respectively of the ranges (x^, ciiO^),
(xiXf, OyO^t (^liCi, (h<h) aJ^e given by
_\fH ^\fh ^Aq/ia
^'•"x;;^,' ^""'y^' ^'^\J^'
Hence log p„ + log p^i = log p,8.
Then if 7 be some numerical constant, real or imaginary, we may define
[c£ Klein, loc. cit.] ^logp^ss the distance between any two points Xi and ^;
where the distance is conceived as a signless quantity, but the ordinary
conventions may hold as to the sign of lengths according to the direction
of measurement. But the definition and the resulting conventions require
further examination accordiug to the different cases, which may arise [c£
subsection (4) below].
(3) Let the point-pair Oi, o^ be called the absolute point-pair. Let
these points be either both real, or let their corresponding co-ordinates,
referred to any real set of reference elements, be conjugate complex numbers.
Then for real points Xi and fii are both real when Oi and o^ are real, and
are conjugate imaginaries when Oi and a, are conjugate imaginary points.
Similarly for X, and fjL^^ and for X, and fi^.
* Cf. PhU. Tram. 1859, and CoUecUd Papen, Vol. ii. No. 158.
t * Ueber die aogenannte Nioht-EnkUdisohe Oeometrie/ Math, Aunalen, Bd. iy. 1871.
862 THEORY OF DISTANCE. [CHAP. I.
Hence when Oi and a, are real, in order that the distance between real
points which both lie on the same intercept between Oi and a, may be
real, 7 must be real When Oi and o^ are conjugate imaginary points, in
order that the distance between real points, such as Wi and x^, may be real,
y must be a pure imaginary ; for ' log pu is a pure imaginary. Let -r be
written for 7 in this case.
(4) Thus if the absolute point-pair, eii and a^, be real, the distance
between any two real points, Xi and a^, lying between them, is defined to
be the real positive quantity ^ log pj2, where 7 is some real number, and
pi2 IB so chosen as to be greater than unity. Since /^u > 1, it follows that,
with the notation of subsection (2),
Ai/i, ^^^^ ^ >-?, and p„ = —^ when — >— .
Assuming — > — , then pi, can be described either as the anharmonic ratio
fh A*»
of the range (^^r,, Ojas), or as the anharmonic ratio of the range (x^, OjOi).
Thus Oi bears the same relation to o^i, as a, bears to /t?,, in this definition of
distance. The points Oi and Xi will be considered as lying on one side of ^,
and the point a, on the other side [cf. § 90]. Let this be called the Hyper-
bolic definition of distance. It is to be noticed that, with this definition a
pair of points, not in the same intercept between aj and a,, have not a real
distance.
If the absolute point-pair be two conjugate imaginary points, the distance
between two real points Xi and os^ is defined to be one of the two values
of —-. log/9i2 which lies between 0 and iry. The ambiguity as to which value
is to be chosen is discussed later in § 204, and its determination is possible in
two ways. Let this be called the Elliptic definition of distance.
(5) Let the limiting case be considered in which the absolute points are
coincident at some point u. Let e and u be the two reference points in the
one-dimensional manifold. Let ai = ai^-h)8it^, a^^ a^ -{■ P^u. Then, when
Oi and Os ultimately coincide with u, a^j^x &nd o^/iS, ultimately vanish. Let
any other points x and y be written, rc = e + fu, y = e + i;K
Now putting A for fiifit^dSu
Aa? = (A - a,f) Oi -09a - aif)aa,
Ay « (A - 0,17) a, -Ox - aii7)a,.
Hence according to Cayley's definition, the distance between x and y is
, ,^ (A-«,g)(/3.-«.i?)
200] cayley's theory of distance. 368
Therefore expanding in powers of a,/A and o^/A and retaining only the
lowest powers, the distance becomes
J7(|-|)(i»-f).
vPa Pi/
/ft n \
Now let 7 increase as ai/)3, and Os/ySs decrease, so that 7 ( ^ ~ -^ ) remains
finite and equal to S, say. Then in the limit when Oi and a, coincide with u,
the distance between the points e + ^i and e + fju is 8(17 — ^). Therefore
this definition of distance is a special limiting case of the more general
definition first explained. Let it be called the Parabolic definition.
200. Klein's Theorem. (1) It can be shown (cf. Klein, loc. cit.)
that this definition of the distance between two points is the only possible
definition, which is consistent with the propositions on congruent ranges in
§ 198 (3) and (4).
Let P>Pi,P2i ••• ai^d pi,pa, Ps, ... be two congruent ranges. Then by
definition the distance ppi = the distance pipa = etc. Also by § 198 (3) and
(4) the ranges are homographic ; therefore [cf. § 71 (1)] the first range can
be transformed into the second by a linear transformation. Let Oi and a, be
the two points on the line which are unaltered by this transformation [cf. § 71
(2)], and firstly assume them to be distinct [cf § 71 (4) and (5)]. Then
[cf. § 71 (8)]
Thus the anharmonic ratio (p„pi„ OiOj) is i/^'*. But the distance between
j>p and j><r is (o- — p) times the distance p^p^, which is any arbitrarily assumed
distance X. Accordingly if Oy and a, be Cayley's absolute point-pair and
|>p= tfoi + ^, />a^^= -^ro, + x'^y ^® obtain
dist.i>pp, = X(cr-p) = j^log^.
But this is the definition of distance already given in § 199 (2), as far as
concerns integral multiples of an arbitrarily assumed length X. But since X
is any length, it may be assumed to be small compared to all lengths which
are the subjects of discourse. Thus the definition must hold for all lengtha
(2) Secondly, let the two points, ai and a,, unaltered by the linear
transformation, be coincident, and write u for either of them [cf. § 71 (7)].
Let e be any other reference point on the line ; then if any point p be
written in the form ^+ fw, it is transformed [cf. § 71 (7) equation D] into
the point «-f(f + S)w, where 8 does not depend on f. Thus if the range
PfPifPi"' be transformed into the congruent range pi, p^fPt-"* the distance
between the points j>p and p^ is (a — p) times the distance between p and pi.
But [cf. § 71 (9y]pi,= e + (i + pS)u(^e + fiu, say), and
Pa = e + {i + aS)u(^e + rfu, say),
w. 23
354 THEORY OF DISTANCE. [CHAP. L
Therefore the distance p^'pp = X (<r — p) = ^ (V ~ ^)» where \ is real. But
this is the Parabolic definition of distance of § 199 (5).
201. Comparison with the Axioms of Distance. The only difficulty
in reconciling the Cayley-Klein theory of distance with the axioms
of § 197 (3) arises from axiom I. For in axiom I. the distance is said to
relate two points of a spatial manifold, whereas the definition of distance of
§ 199 relates four points of the manifold, namely the two points of which the
distance is defined and the two points forming the absolute. But the two
points which form the absolute, if real, are at an infinite distance from every
point of the spatial manifold. They may be considered as extreme, or
limiting points, of the manifold. Thus the distance only relates two points
arbitrarily chosen* Again if the absolute point-pair be imaginary, and the
distance only relates real arbitrary points, the other points which enter
into the definition are special points and are imaginary.
202. Spatial Manifolds of many dimensions. (1) Consider a
spatial manifold of i^ — 1 dimensions, where v>% Assume that Cayley's
definition of distance applies to eveiy straight line in it.
Let the whole, or part, of the spatial manifold be such that Cayley's
definition of distance, in the same one of its three forms, applies to any two
real points in it ; so that a real distance exists between them. Then such a
manifold, or such a part of a manifold, will be called a Space of i/ — 1
dimensions. If the Space of i^ — 1 dimensions be not the complete spatial
manifold, then there must not be a real distance between any point in Space
and any point in the remaining part of the spatial manifold. Let the remain-
ing real part of the spatial manifold be called Anti-spaoe. Thus a spatial
manifold is either such that its complete real portion forms Space ; or it is
such that its complete real portion is partly Space and partly Anti-space.
(2) Consider any triangle oho in the complete spatial manifold, the
whole or some part of which forms the space considered. Let h and c be
real, and a either real or imaginary. Let the distance between h and c be
real and finite.
It follows from § 197, Axiom III., that if a be one of the points of the
absolute point-pair of the line ob, it must also be one of the points of the
absolute point-pair of the line oc. Hence all the points which form the
absolute point-pairs of all straight lines must form either an entirely imagin-
ary sur&ce, or a closed surface; and the part of the spatial manifold,
which forms Space, must lie within the surface. For [cf. § 82 (1)], when
the absolute is real, every straight line through any point in space must
cut the absolute in a pair of real points. Then the part of the manifold
outside the closed surface is Anti-space.
201—203] SPATIAL MANIFOLDS OP MANY DIMENSIONS. 355
(3) Also every straight line, containing points in the spatial manifold,
must cut this surface, whether it be real or imaginary, in one point-pair.
The only algebraic surface for which this is possible is a quadric.
Let it be assumed in future that the absolute point-pairs form a quadric,
which is either entirely imaginary or real and closed. Let this quadric be
called the Absolute.
(4) When the absolute is imaginary, the spatial manifold is called
elliptic*. There are two forms of elliptic geometry ; the polar form in which
the symbols +^ and ^x represent the same point at opposite intensities
[cf. § 89 (1)] ; the antipodal form in which + x and — x represent different
points [c£ § 89 (2)]. The discrimination between the two forms was first
made by Klein.
When the absolute is real and closed, the spatial manifold is called
hyperbolic. In hyperbolic geometry the symbols 4- x and — x represent the
same point at opposite intensities.
Parabolic Space is a special limiting form which both Elliptic and
Hyperbolic Space can assume, when the absolute degenerates into two
coincident planes [cf. § 212 below].
(5) Let the distance between the two points a and b be written D (ab)
as an abbreviation for ' distance ab.'
203. Division of Space. In the polar form of elliptic geometry a
plane does not divide space. For if x and y be any two points and L any
plane, the straight line osy cuts the plane L in one point p only. But
[cf. 90 (5)] there are two intercepts between x and y. Thus the plane L cuts
one of the intercepts and does not cut the other. Hence it is always possible
to join any two points by an intercept of a straight line which does not cut
a given plane.
(2) But two planes do divide space. For it is possible to find two
points, such that each of the two intercepts joining them cuts one of the two
planes. For any straight line must cut the two planes in two points, say in
p and q : on the straight line pq take two points x and y, one on each of the
two intercepts joining p and q. Then the planes divide x from y in the way
stated.
(3) In the antipodal form of elliptic geometry a plane does divide space.
For any straight line cuts a plane L in two antipodal points. Then if one
intercept between two points x and y contains only one point p on the plane,
the other intercept must contain the antipodal point. Thus x and y are
divided from each other by the plane. Points x and y, which are not divided
* Klein confines the term Elliptic to the Polar form of EUiptic Geometry. The Antipodal
form 18 called hy him Spherical Geometry.
23—2
356 THEORY OF DISTANCE. [CHAP. I.
from each other by the plane, must be such that one intercept between
X and y does not cut the plane and the other intercept contains the two
antipodal points of section, namely ±p,
(4) In the hyperbolic geometry a plane does divide space. It might
have been wrongly anticipated, since ±x represent the same point, that
results analogous to those in the polar form hold. But in elliptic geometry
space is the whole of the real part of the positional manifold ; whereas in
hyperbolic geometry space is only the part of the positional manifold within
the closed absolute. Now no straight line lies completely within the absolute.
Accordingly if one intercept, joining two points in space, itself lie completely
in space, the other intercept passes out of space. Hence, ignoring points
outside space, points joined by an intercept, lying entirely in space and cut
by a plane, are divided fix)m each other by that plane.
204. Elliptic Space. (1) Let the absolute be imaginary, so that the
space is elliptic : let it be chosen to be the self-normal quadric [c£ Bk. IV.,
Ch. III.]. Then its equation can be written (x\ai) = 0. Also let it always be
assumed that, when x represents any real point, (x \ x) is positive. Then from
§ 199 (4) and equation (2) of § 123 (9), the distance between any two points
Xi and x^ is
where the inverse trigonometrical functions are to denote angles between
0 and ir.
(2) If X and —x represent the same point [cf. § 89 (1)], the am-
biguity of sign must be determined so that ±(aa|a!a) is positive; for this
choice makes the distance of a point from itself to be zero. Hence, in the
polar form of elliptic space, D{xiX^ is not greater than ^tt/.
(3) If X and — x represent different points [cf. § 89 (2)], then the upper
sign is to be chosen in determining the ambiguity. Thus if (2 be the distance
between x and y, and d! the distance between — x and y,
cos-=— ^^M__ and coo ^'- ^"^'^^ - "^^1^^
Hence, in the antipodal form of elliptic space, D{xy) is not greater
than Try.
206. Polar Form. (1) It is necessary, for the elucidation of the
distance formula of the polar form of elliptic space, to investigate the circum-
stances under which {x \z) and {y \z) are of the same and of opposite signa
Let the polar plane of x with respect to the absolute cut xy in x\ and let
that of y cut xy in y'. Let the closed [cf. § 65 (9)] oval line xyxj/ of the
i
204, 205] POLAR FORM. 357
figure represent the complete straight line xy. Any point z on this line can
be written in the form cue + ^af. For the sake of simplicity assume that a is
positive and does not change, and that f alone varies as z shifts its position on
Fia. 1.
the line. Then for one of the two intercepts between x and x\ f is positive ;
for the other, f is negative ; let xyx' be the f positive intercept. Assume
y = ax'{- paf, thus ^ is positive. Also we can write y' = /9 (a?' |aj') a? — a (a: | x)af
thus j/ is on the f negative intercept. Also let it be noted that with the
assumed form of y, {x\y) \=a(x |a?)] is positive.
Now -ar=ow?+fa?' = V{a«(a;|a;)4-fi8(a?'|a?')}y + V(a)8-af)y',
where X-* = a« {x\x)'{' ^{x'\af).
Accordingly {x \z) is positive at all points of xy. And
(y 1^) = X» {a» {x\x) + f/3 {x \af)] (y |y).
Hence, remembering that as z moves from a/ to y^ in the direction of the
0^ ix \icS
arrow f changes gradually from — oo to — ^ /.' > , we deduce that (y |^) is
p yx \x)
positive when 2: is on the intercept afyxi/ between of and y', and is negative
when 2r is on the other intercept.
(2) Secondly let z be any point not necessarily on the line xy. Now
[cf. § 72 (5)] z can always be written in the form z^ + j>, where Zi is on the
line xy and p is on the subplane which is the intersection of the polars
of X and y; and this representation is possible in one way only. Then
{x\z)^{x\z^\ and {y\»)^(y\zi).
(3) Hence, summing up the results of (1) and (2), we see that if z be
separated from x and y by the polar planes of x and y, then {x \z) and (y \z)
are necessarily of different signs, provided that {x\y) is positive. But if z
be not separated from x and y by the polar planes, then {x \z) and (y \z) are
necessarily of the same sign, when (x\y) is positive Thus if (y\z\ {z\x)y
{x \y) are all of the same sign, they are all positive.
358 THEORY OF DISTANCE. [CHAP. 1.
(4) Let the intercept between x and y on which af and y' do not lie be
called the intercept, while that intercept on which of and j/ do lie is called
the polar intercept.
206. Length of Intercepts in Polar Form. (1) If a?, y, ^ be three
collinear points, it is as yet ambiguous as to which lies between the other
two, since the straight line is a closed curve. The definition of distance has
however really decided the question, as is shown by the following inves-
tigation.
(2) Let {os\y) be positive, and firstly let z lie on the intercept [cf. § 205
(4)] between x and y.
Put z^\X'\' iiy\ then X, /Lt, {x\z) and {y\z) may be assumed to be
positive.
Hence
D(xz)_ (x\z) . D(xz)__ I {xz\xz) __ / (xy\xy)
cos— - ;/{(,^(7|^y} ' ^^~;f ~v (^|iH^)"''*v (^k)('^k)'
and cos^-^^ = ^yl^> sin^^^ = X Ai^d^).
7 VKy \y) {^ k)} ' 7 V (a? \x) (z \z) '
Thus ,i^^(^^)+^(^y)^M(yk)|v(^k) / (^^^^^^^^
Also cofl ^ ^^^) "^ ^ (^y^ ^ ^^ 1^^ ^y !^) "^ ^^ ^^^ i^y^ ^ (^ k) (y k) + (^y 1^)
7 (-2^ k) V{(a? 1^) (y !y)} {^\^) V{(^ |a?) (y jy)}
V{(«l«)(y|y)) 7
Hence D (a?^) + D (^y) = D (ojy).
Thus when z lies on the intercept between x "and y, as defined in
§ 206 (4), z lies between x and y according to the meaning of § 197,
axiom II.
(3) If z lie on the intercept between y and x', then y lies on the
intercept between x and z. Thus from subsection (2),
D{xy) + D(yz) = D(xz).
Similarly if z lie on the intercept between x and y^, then a? lies on the
intercept between y and z, and
2)(ya?) + i)(a?xr) = 2)(y2r).
(4) If z lie between a/ and y', then / lies on the intercept between
X and y, and each of the points x, y, z is separated from remaining two by
the pair of polar planes of those two ; so that each point lies on the polar
intercept of the other two. Assume (x\z) positive and (y\z) negative: also
let z — Xx — fiy, where X and /a are positive [cf. § 205 (1)].
a^
206] LENGTH OF INTERCEPTS IN POLAR FORM. 369
rjy, D{xz) («!-?) . DCxz) I (xu\xy)
Then cos -^^-^ = .., \ ;/ , .. , sm —5^^= fi./, ^ \\^/ . ,
7 VK^kX^k)}' 7 v(y|y)(^k)'
Hence
•
7 (y|y) V ('^kX'^k) (y|y) V («k)(^|.
)(-^k)
(^k) 7
Also
cos -^"^^^ +-P(y-^) ^ -(^ly)(yk)-^(^yl^y) ^ -(^ly)(yk)"(^yl^)
7 (y ly) V{^ k) (^ k)} (y |y) V{(a? k) (^ k)}
^ "(ylyX^k) ^ coo^^"^^^
(y|y)V{(^k)(^k)} 7
Hence D (ajy) + D (y^) = tt/ — 2) (a?-?),
or D (yz) + D (zx) -{- D (xy) == Try (A).
Hence no one of the points x, y or z lies between the other two according
to the meaning of § 197, Axiom II.
(5) This difficulty in the reconciliation of the Polar form of Elliptic
Geometry to the Axioms of Distance may be obviated as follows. The
distance between two points must be specially associated with the intercept
between them ; since for the intercept only is the axiom II. of § 197 true.
Let the distance between two points be also called the length of the
intercept. Thus the intercept itself is considered as possessing a quantity of
length.
(6) Again the polar intercept may also be considered as possessing a
quantity of length. For, since (x\a/) - 0,
D (xx) = 7 COS"* 0 = ^-TPy.
Also and similarly
D{ya/) = D(xi/)r.^^^D(xy).
Hence D {x'j/) = D {xx') - D (xt/) = D (xy).
Thus D(ya/) + D(xy)'\-D(i/x)^wr-'D(xy).
Hence wy — D (xy) may be considered as the length of the polar intercept
between x and y, since it is the sum of the lengths of its three parts.
Accordingly the whole length of the straight line may be considered to
be Try, This also agrees with equation (A) of subsection (4).
(7) The paradox of subsection (4) can now be explained. For each of
the three points lies on the polar intercept between the other two : and the
sum of the distances of any two from the third is in each case equal to the
360 tflEORY OF l>lStANCa. [CHAP. L
length of the polar intercept. Thus axiom 11. of § 197 ought to be amended
into, the sum of the lengths of the parts which make up either the intercept,
or the polar intercept, is equal to the length of the intercept, or of the polar
intercept, as the case may be.
(8) • Also if 7 cos-^ /f/ . w 1 XI gives the length of the intercept
^ V{(^k)(yly)}
between x and y, then 7 cos~^ /f 1 \/ 1 v ^^^ ^^^ length of the polar
»s/[x \x) yy \y)\
intercept ; and vice versa.
Let that intercept between x and y of which the length is
COS"^
{^\y)
be called the intercept (a? |y), or ^; and let the length of the intercept {x \ y)
be called xy. This name is useful in the ordinary case in which it is unknown
and immaterial whether {x\y) is positive or negative. If (a?|y) be positive, |
the intercept {x\y) is the intercept between x and y according to § 205 (4).
(9) It is necessary in this connection to distinguish carefully between
the points x and y, and the terms x and y by which they are symbolized
[c£ § 14, Definition^ All congruent terms [c£ § 64 (2)] denote the same
point (or regional element). Two points x and y divide the complete
straight line into two intercepts. The sum of the lengths of the two
intercepts is Try. The length of the shortest intercept, which is the distance
between the points x and y, is D(xy). The length of the other (polar)
intercept is 7r^''D{xy). The terms x and y, written in the form {x\y) or
o^y define one of these intercepts. If {x\y) be positive, this intercept is the
intercept, and is of length xy = D(xy), If {x\y) be negative, this intercept
is the polar intercept, and is of length xy^7ry — D{xy), Let a/' = — a;,
y" = — y. Then the terms a/' and y" denote the same points as x and y. Also
(a/^ ly") = (^ |y)> hence the intercept (a/' |y") is the same as the intercept (x |y).
But the intercepts (a?"|y) and (a;|y"), which are the same intercept, are
always the other intercept to the intercept (x \y) or (a/' |y").
Thus, summarizing and repeating the distinctions between D (xy) and ^ ;
D(xy)^DixY) = D(^''y) = D{^r>
^=^y'; W=^'y'y ^+^y='r7;
xy^D {xy\ if (a? \y) be positive ;
a/'y = D {xy), if (a? I y) be negative ;
^(^)<i^7; ^<'r7.
Also the length of the intercept (a; |y) is written ^. Then Wy (as well
as (a;|y)) may also be taken as this name of the intercept. It is not often
of much importance to know whether i»y = D {xy) or Try — D {xy).
I
207] LENGTH OF INTERCEPTS OF POLAR FORM. 361
(10) If z be the point x + fy, then when f is positive z lies on the
intercept {p\y\
For cos^ = (^k) + f(^ly)
Hence as f changes gradually from 0 to + oo , cos — diminishes gradually
from 1 to -,17— T-TTivr • and this whether ix \v) be positive or negative.
sl{{x\x)(\i\y)\ \ \9J r 6
Thus ^ gradually increases from 0 to o^. Similarly at the same time
zy gradually decreases from ^ to 0. Hence z must lie in the intercept
(^ |y).
207. Antipodal form. (1) In the antipodal form of elliptic geometry
the intercept between x and y is that intercept which does not contain the
Fio. 2.
antipodal points — x and — y ; the intercept containing the antipodal points
is called [cf § 90 (6)] the antipodal intercept.
(2) Now by a proof similar to that in the previous article, if z lie in the
intercept between x and y, D (xz) + D (zx) = D {xy). If z lie in the intercept
between y and — a?, D (xy) + D {yz) = D {xz). If z lie in the intercept between
X and —y^D {yx) + D (xz) = D (yz).
(3) If z lie in the intercept between — x and — y, let / (= — ^) be the
antipodal point to z. Then / lies in the intercept between x and y.
Hence by subsection (1)
D(xjO + D(z'y) = D(xy).
But by § 204 (3),
D(xz) + D(x:^) =:'jrf = D(yz) + D(y:^).
Hence D (xz) + D (zy) = 27r7 — D (xy).
Also
D (xy) + D (yz) « D (xy) + wy - i) (y/) ^iry + D (xsT) = 27ry - 1) (xz),
and i) (yx) + D (a?^) = 27r7 - D (yz).
362 THEORY OF DISTANCE. [CHAP. I.
Thus no one of the three points x, y, g lies between the other two in the
sense of axiom II. § 197. Accordingly this axiom is not literally satisfied ;
however the following explanations and additions shew that it is substantially
satisBed.
(4) Analogously to the similar case of the polar form, let the distance
between x and y be called the length of the intercept between x and y.
Then the length of the intercept between y and — a; is tt^ — D (xy), and this ^
is also the length of the intercept between — y and x. The length of the
intercept between — x and — y is D {xy). Hence adding the three parts, the
length of the antipodal intercept between x and y is 2iry — D (xy).
Thus the length of the whole straight line is 2wy.
(5) The paradox of subsection (3) can now be explained. For each
of the three points lies on the antipodal intercept between the other two:
and the sum of the distances of any two from the third is in each case equal
to the length of the antipodal intercept. Thus axiom II. § 197 ought to be
amended into, The sum of the lengths of the parts which make up either
the intercept or the antipodal intercept is equal to the length of the •
intercept or of the antipodal intercept, as the case may be.
208. Hyperbolic Space. (1) Secondly let the absolute quadric be
real and closed. Then from § 199 (4) and equation (3) of § 123 (10), the
distance D (xy) between any two points x, and y within the quadric is
-0 (xy) = i7log pu = 7C0sh-^ /f/"! w i x) = 7 sinh-^ . / ^ ■ w i v »
V y/ T/ 6f« J ^{(x\x)(y\y)} ^ V(^k)(y|y)
The ambiguity of sign must be determined so that ±(x\y) is positive.
It has been proved in § 82 (9) that (x \x) and (y |y) are of the same sign :
hence {(«|a?)(y]y)} is necessarily positive.
(2) The test as to which sign of the ambiguity is to be chosen is
derived from the following lemma ; which, it is useful to notice, applies to
any closed quadric (a][^) = 0, and not solely to the absolute in its character
of self-supplementary quadric.
Let e, X, y, z be four points within the quadric. Then, if (e \x\ (e |y), (e \z)
are of one sign, also (y \z\ (z \x\ (x \ y) are of one sign.
For let the line xy cut the polar of e in e', and let xz cut it in e'\ Then
we may write y = Xa? + i;e', z = fix + ^e". Hence, since (^ |tf')r=0 = (e|e"),
(e\y) = X(e\x) and (e\z)=fi(e\x). Therefore from the hypothesis X and fi \
are positive.
Again as 17 varies between — 00 and + 00 , y takes all the positions on the
line xe\ Also (x\y) = \(x\x) + i](€^ \x). Hence (x \y) is a linear function of
the variable 17 ; and thus as rj varies, (x y) can only change sign when it
208, 209] H7PERB0LIC SPACE. 363
vanishes or is infinite. But when (x\y) vanishes, y must lie on the polar
plane of x, and this plane is entirely outside the quadric [of. § 82 (6)];
similarly when (x \y) is infinite, tf is infinite and y coincides with e' which is
outside the quadric since it lies on the polar plane of e.
Thus for all points y on that part of the line xe which lies within the
quadric, {x\y) has the same sign. Now put 17 = 0. Hence {x\y) has the
same sign as \(x\x). But X is positive. Thus {x\y) has the same sign
as (x \x). Also {x \x) has the same sign for all points within the quadric, say
the positive sign. Hence (x\y) is also positive. Thus the proposition
is proved.
(3) Let (x\x) be always assumed to be positive for points within the
quadric : also let a point x within the quadric be said to be of standard sign
when (e x) is positive, where e is any given point within the quadric chosen
as a standard of reference. Then it follows from the above that for all points
of standard sign within the quadric, {x \y) is positive.
Thus the distance between two points x and y, within the quadric and
of standard sign, is
D (xy) = i7 log pn = y cosh-^ -,,y (^ ly^ - ,. = 7 sinh-^ /rf?J?l •
In future all symbols arbitrarily assumed to represent points within a
real closed absolute will be assumed to represent them at standard sign.
(4) In hyperbolic space there is only one intercept between two points
X and y which lies entirely within the space. Also if z lie within this
intercept
D(xz)-\-D(zy)^D(xy),
Hence there is no ambiguity as to the application of axiom II. of § 197.
The distance between x and y will be called the length of the intercept
between x and y.
The distance of any point from any point on the absolute is infinite.
Thus the length of the part of any straight line within the spatial manifold
is infinite.
209. The Space Constant. It is formally possible to assume that
7, instead of being an absolute constant, is constant only for each straight
line ; and accordingly is a function of any quantities which define the special
straight line on which Xi and a^ lie. Such quantities can necessarily be
expressed in terms of the co-ordinates of Xi and x^, since these points define
the line XiX^. Hence the assumption of 7 as a fiinction of the co-ordinates of
the straight line joining the points does not appear necessarily to offend
against the axioms of § 197. Let the assumption be made that 7 is constant
and the same for all lines. Let 7 be called the space-constant.
364 theory of distance. [chap. i.
210. Law of Intensity in Elliptic and Hyperbolic Geometry.
(1) The law of intensity (c£ Bk. III. ch. iv.) is, also settled, if the
assumption* be made that, when x^ and x^ are of the same intensity, Xi'\-x^
bisects the distance between Xi and x^ ; where for the polar form of elliptic
geometry {x^ \x^ is assumed to be positive, and for hyperbolic geometry
x^ and x^ are both of standard sign. No special explanation is required for
the antipodal form of elliptic geometry, since o^i + ^ is to bisect the distance
between a?, and x^y and a^ — a?, is to bisect the distance between x^ and — Xi.
Then by §§ 204 and 208,
[(Ci\{Xi-\-x^] ^ {a?al(a?i+a?a)}
Hence y(x^ [a?,) - ^{x^ {x^)} y(xj \ a?0 (x^ |a?a) - (x^ Ix^)] = 0.
Therefore either (xi \xi) = (x^ {x^), or (xi \xi) (x2 |a?j) — (xi {x^y = 0.
The second alternative is equivalent to {XiX^ \xiX^ = 0. This implies that
the line x^x^ touches the absolute [cf. § 123 (5)] ; and this presupposes
special positions for a^ and x^. In fact for such a case in elliptic geometry
the line x^x^ would then be imaginary ; and in hyperbolic geometry x^ and x^
would lie outside the absolute.
Hence the alternative, {xi\x^ = {x^\x^, must be adopted. Accordingly
if the point x has a given intensity, {x \x) is independent of the position
of X. Thus with a proper choice of constants the intensity of a; is *s/{x \x) ;
so that {x \x) = 1, when x is at unit intensity.
(2) Then, if Xi and ^ be at unit intensity and (xi\x2) be positive
(except for antipodal elliptic space), the formulae for the distance between
them become,
and
dm = ^. log pu = 7 cos-^ (a?i \x^) = y sin-» ^/(xi^% \xi^) ;
^ = 2 ^og pia = 7 cosh-^ (xi |aj,) = 7 sinh"^ V(- ^i^i 1^^);
acxsording as the space is elliptic space or is hyperbolic space (of any number
of dimensions), where in both cases x^ and x^ fulfil the condition
(^l«i) = l=(^k).
(3) As an illustration of these formulae consider antipodal elliptic space
of two dimensions.
Let the absolute be (x\x)^ fi* + f»* + f,* = 0.
Then the conditions, {x |a;) » 1 = (y \y), become
* This Bssompiion is made by Homershftm Cox, loe. cit.
210, 211] LAW OF INTENSITT IN ELLIPTIC AND HYPERBOLIC GEOMETRY. 365
And cos - = (a: \y) = f jiy, + ^tV^ + fji/s,
sin - = ^/{xy \xy) = V{(f s^/j - izV^^ + (fsi/i - f ii/j)* + (f i^/a - f ii/i)"}.
These are the formulae of the ordinary Euclidean geometry of a sphere ;
where fi, fji fs and i/j, i/ai ^j are direction cosines.
211. Distances of Planes and of Subreoions. (1) As yet only
the distance between points has been defined. The same principles can
easily be applied to planes.
For any planes X and T can be expressed in terms of their polar points
with respect to the absolute. Thus X^\xy and F = | y. Hence, if the
absolute be imaginary, {X \ X) and {Y\Y) are necessarily of the same sign.
If the absolute be real and closed, {X \ X) and {Y\Y) are of the same sign,
when X and y are either both within or both without the absolute. If x lie
within the real closed absolute, the plane X contains [cf. § 82 (6)] no points
lying in space, but only points in anti-space ; but if ^ lie without the
absolute, then [c£ § 82 (7)] the plane X contains points in space as well as
points in anti-space.
Let X and T be any two planes, and suppose that the plane XX + ^F
touches the absolute quadric.
Then Xjfi must be one of the two roots Xi//Lh and \^(h of the equation
X« (-X' l-Sf) + 2X/i (Z I F) + /i» (F| F) = 0.
Let A^ and A^ be these two tangent planes ; then the anharmonic ratio
of the range {XF, AiA^ is Xi/i^X^, and this ratio is either real or of the
form c***, where ^ is real. Let it be called p.
Then if p be real, the measure of the separation between X and F can
be defined to be ^ log p ; and if p be imaginary, it can be defined to be
^. log p ; where k and k' are constants.
There is no reason why either k or k' should necessarily be equal to the
' space-constant ' 7. But there is no real loss of generality, and there is a
gain in the interest of the analogy to ordinary geometry, if /c = 7, and k =■ 1.
For it will be found that the hyperbolic measure of separation between
planes can then be identified with the distance between two points; and
the elliptic measure of separation can be considered as the angle between
them, which is of no dimensions in length.
(2) Thus, [cf. § 124], it follows that the separation between two planes
X and F is that angle between 0 and tt given by
/i-Ii - ±(Z|F) . _, / (ZF|ZF)
^-2,%gp-^co& ^{(XIXXFIF)}""^^^ V {(Z|Z)(F|F)} '
when {^\yy< (-T |X) (F| Y).
366 THEORY OF DISTANCE. [CHAP. I.
And the separation is
d = |logp = 7COsh i^-^__-^_-^ = ^sinh ' ^ ^^^x \X) (¥{¥)}'
when (Z|F)»>(Z|Z)(F|F).
It must be noticed that the distinction between these two cases must not
be identified simply with that between Elliptic and Hyperbolic Geometry as
defined above. The trigonometrical functions must however always be
adopted in Elliptic Geometry. This question will be considered in the
succeeding chapters as far as it concerns Hyperbolic Geometry.
(3) Furthermore the ambiguity of sign is capable of being determined
by exactly the same methods as obtained for points. But with respect to
planes, in order to obtain an interesting extension of the ideas of ordinary
geometry, the ' polar ' form is invariably adopted, namely, + X and — X are
considered as representing the same plane at opposite intensities.
(4) If the elliptic measure of distance between planes has to be adopted,
the measure of separation of planes is called the angle between them.
The ambiguity of sign in the formula for the cosine of the angle leads to
the definition that planes make two supplemental angles with each other,
^andTT — ^; and that of the two the acute angle is the measure of the
separation of the planes.
(5) The law of intensity of planar elements is determined by the same
principles as that of points. Let it be assumed that if X and F be planar
elements of the same sign and at the same intensity, then X + F bisects the
distance between X and F. Hence the defining equation of unit intensity
can be written, (X\X) = S, where 8 is a constant which will be determined
later separately for Elliptic and Hyperbolic Geometry according to con-
venience. In Elliptic Geometry S is always of the same sign : let it there-
fore be chosen to be unity. In Hyperbolic Geometry it is convenient to
choose S to be positive or negative according as the (real) plane does or does
not cut the absolute : let it therefore be chosen to be ± 1.
(6) It is in general impossible to define one single measure of separation
between any two subregions X, and F^, of o- — 1 dimensions. But if they
are both contained in the same subregion of a dimensions, then considering
the latter subregion as the complete region, X^ and F^ have the properties
of planes in regard to it. Also the absolute in this complete region may be
taken to be the section of the absolute by the region.
Hence in this case, c£ § 124 (4), the measure of the separation of X^ and
Fa (with the conventions, already explained, determining ambiguities) is
either
± (X, I Fe) ,_,_ ±{X,\Y,)
^^ TfWXKPTTF;;)} ' '''' '^ VKX|Z.)(F.|F4-
211, 212] DISTANCES OF PLANES AND OF SUBREGIONS. 367
Definition. |Z, and |F, are called the absolute polar regions of X,
and Y^.
It is obvious that the separation between two regions is equa] to that
between their absolute polar regions.
212. Parabolic Geometry. (1) If the parabolic definition of distance
hold for every straight line, then every straight line must meet the absolute
in two coincident points. Hence the absolute must be two coincident planes.
It can be seen as follows that the elliptic and hyperbolic definitions for i/ — 1
dimensions both degenerate into the parabolic definition, when the absolute
is conceived as transforming itself gradually into two coincident planes.
(2) Let the co-ordinate points Ci, C2> ••• ^k be v self-normal points, then
the equation of the absolute takes the form,
Now conceive the form of the quadric to be gradually modified by
Oa, ... a^ diminishing, till they ultimately vanish, while a^ remains finite.
Then ultimately the equation of the quadric becomes o^^ = 0 ; that is to
say, the quadric becomes two coincident planes, the equation of each plane
being fj = 0. Also the i/ — 1 co-ordinate points ^9,^3, ... e^ lie in this plane,
and the point Bi without it.
Also, cf. § 1 23 (6), {xy \xy) = 2a,a, (fpiy, - f.iyp)'.
Assume that, as the quadric approaches its degenerate form,
7 7 7
where the ks are finite and 7 is ultimately infinite.
Then ultimately,
7 T 7
Similarly {x\x) = a^f i», (y \y) = airfi\
Then if the geometry be elliptic and 7 be the space-constant,
V (af\x)(y\y) 7«ifi^i «! fi^i
Now, since the geometry is elliptic, a^ and k,2, k^, ... k^ are all of the same
sign. Put ^ = i8p».
Hence d = 2^8^' itbL^MT .
If the geometry be hyperbolic,
1
368 THEORY OF DISTANCE. [CHAP. I.
Now, since the geometry is hyperbolic, the absolute is a real closed
quadric; and hence [of. § 82 (5)] ai must have one sign and /cs, k^, ... x,
another sign. Put — = — ^p".
Hence d^lfi/^P^^\
(3) Thus as a limiting case both of Elliptic and Hyperbolic Geometry,
we find a space with the distance between any two elements given by
where the v — 1 co-ordinate elements e,, ^s» ••• ^i^ ^^6 on the absolute plane at
an infinite distance.
213. Law op Intensity in Parabolic Geometry. (1) Let ei be the
reference element not in the absolute plane, and let u^, ti„...u,, be the
reference elements in the absolute plane. Let it be assumed, as in § 210 (1),
that, when x and y are of the same intensity, x-i-y bisects the distance
between x and y.
Now let X = f 1^1 + Sfw, y = Vi^ + Siyi^ ; then
« + y = (fi + %)«! + 2 (f + i?)w.
Also the distance between x and a; + y is by § 212 (3)
Similarly the distance between x + y and y is
Vi (f 1 + Vi)
Hence since these distances are equal, f i (f i + Vi) = ^i (f i + Vi)f ^^^ thence,
(2) Hence the intensity of the point a; is a function of ^i only ; but by
§ 85 (2) it must be a homogeneous function of the first degree. Thus the
intensity of x is X^i, where X is some constant ; and, if Ci be chosen to be at
unit intensity, then X^l. Hence the absolute plane is the locus of zero
intensity and the law of intensity explained in § 87 (4) must hold. And the
expression for a point x at unit intensity is ^ + S^, where ei is a^ unit
intensity.
Also the distance between the two points ei + S^ and ei + Xt^u, both at
unit intensity, is 2^8^* (f — rjy.
Furthermore by properly choosing the intensities of ti^, ti^, ...u^, this
expression for the distance can be reduced to 2(f — i;)*. Thus* parabolic
* Cf. Biemann, Ueher die Hypothe$en^ welehe der GeometrU »u Grunde liegeUy Collected
Mathematicftl Works.
213] LAW OF INTENSITY IN PARABOLIC GEOMETRY. 369
space of V — 1 dimensions can be interpreted to be simply an ordinary
Euclidean space of that number of dimensions; where CiU^, Cit^s, ...^i^k are
V— 1 axes at right-angles, and fa* fs'-fi^ ar© rectangular Cartesian co-
ordinates. The interpretation of (the vectors) n^, w,, ... u^ will be considered
in Book VIL
Historical Note. An interesting critical * Short History of Metageometry' is to be
found in Chapter I. of The Foundations of Oeometty, by Bertrand A. W. Russell,
Cambridge, 1897. Klein also gives an invaluable short history of the subject in his
lithographed Vorlesungen iiber Nicht-Eukliditche Qeometriey GOttingen, 1893 ; he makes
the important division of the subject into three periods. The following are the creative
works of the ideas of the three periods.
First Period,
Lobatschewsky, Qeometrische UTttersuchungen zur Theorie der Parallel-linieny Berlin,
1840; translated by Prof. G. B. Halsted, Austin, Texas, 1891. Lobatschewsky's first
publication of his discovery was in a discourse at Kasan, 1826 (cf. Halsted's preface);
and subsequently in papers (Russian) published at Easan between 1829 and 1830 (cited
by Stackel and Engel, cf. below).
John Bolyai, The Science Absolute of Space^ 1832 ; translated by Prof. Halsted, 1891 ;
also cf. German edition by Frischauf, cited below. The original is written in Latin, and
is an appendix to a work on Geometry by his father, Wolfgang Bolyai.
Second Period,
Rieman, Ueber die Hypothesen^ welche der Geometric zu Orunde liegen, written 1854,
Gesammelte Werke ; translated by Clifford, cf. his Collected Mathematical Papers.
Helmholtz, Ueber die thatsHchlichen Orundlagen der Oeomeirie, 1866, and Ueber die
ThcUsacher^ die der Oeometrie zum Orunde liegen, 1868; both in the Wissenschaftliehe
Abhandlungen, Vol. ii.
Beltrami, Saggio di Interpretazione della Geometria nxm-EucHdea^ Giornale di
Matematiche, Vol. vi. 1868 ; translated into French by J. Hoiiel in the AnndUs
Scientifques deV Scale Normale Sup&ieure, Vol. vi. 1869.
Third Period.
Cayley, Sixth Memoir upon QuanticSy Phil. Trans., 1859 ; and. Collected Papers^
Vol. n.. No. 158.
Klein, Ueber die sogenannte Nicht-Euklidische Geometric^ two papers, 1871, 1872,
Math. Annalen^ Vols, iv., vi.
Lindemann, Mechanik bei Projectiven Maasbestimmung^ 1873, Math. Anncden, Vol. vii.
Lie, Ueber die Grundlagen der Oeometrie^ Leipziger Berichte, 1890.
A bibliography up to 1878 is given by G. B. Halsted, American Journal of Mathematics,
Vols, "t., n.
The following very incomplete list of a few out of the large number of writers on the
subject may be useful :
Flye, Ste Marie, Stvdes analytiques sur la theorie des paraUMes, Paris, 1871.
M. L. Gerard, Thfese, Sw la G^ometrie Non-EuclidiennCy Paris, 1892.
Poincar^ Theorie des Groupes Fuchsienties, Acta Mathematica, Vol. i., 1882.
Clebech and Lindemann, Vorlesungen iiber Geometricy VoL ii. Dritte Abtheilung,
Leipzig, 1891.
Frischauf, Elemente der Absoluten Geometric Tiach Johann Bolyai, Leipzig, 1876.
w. 24
870 THEORY OF DISTANCE. [CHAP. I.
Killing, Die Nichi'Euhlidischen Raumformen in Analyti$cher Bekandlung, Leipzig, 1885.
Stackel and Engel, Die Tkeorie der ParaUel-linien von EvJclid his auf OausSy Leipzig^
1896. This book contains a very useful bibliography of books on the Theory of Parallels
from the year 1482 to the year 1837.
Veronese, cf. loc. cit. p. 161.
Bumside, On the Kinematics of Non-EiLclidean Space^ Proc. of Lond. Math. Soc., 1894.
Clifford, Preliminary Sketch of Biquatemions^ Proc. of Lond. Math. Soc., 1873, and
Collected Mathem^cUioal Papers,
Newcomb, Elementary Theorems relating to the Geometry of a spa/se of three dimensions
and of uniform positive curvatttre in the fourth dimensiony Crelle, Vol. 33, 1877.
The philosophical questions suggested by the subject are considered by Russell,
Foundatiofis of Geometry (mentioned above); in this work references will be found to
the previous philosophical writers on the subject.
The first application of an * extraordinary ' algebra to non-EucUdean Geometry was
made for Elliptic Space by Clifford, Sketch of Biquatemions, Proc, of London Math,
Society^ Vol. iv. 1873, also reprinted in his Collected Papers ; this algebra will be con-
sidered in Vol. II. of this work. The first applications of Grassmann*s Calculus of
Extension to Non-Euclidean Geometry were made independently, by Homersham Cox
(cf. loc. cit. p. 346), to Hyperbolic and Elliptic Space, and by Buchheim to Elliptic Space ;
On the Theory of Screufs in Elliptic Space^ Proc, London Math, Soc., 1884 and 1886,
Vols. XV. XVI. xvn.
The idea of starting a ^pure' Metrical Geometry with a series of definitions referring
to a Positional Manifold is obscurely present in Cayley's Sixth Memoir on Quantics; it
is explicitly worked out by Homersham Cox (loc, cit.) and by Sir R. S. Ball, On the Theory
of Content y Trans, of Roy. Irish Academy, Vol. xxxx. 1889. Sir R. S. Ball confines himself
to three dimensions, and uses Grassmann's idea of the addition of points, but uses none of
Grassmann's formuke for multiplication. But the general idea of a pure science of
extension, founded upon conventional definitions, which shall include as a special case
the geometry of ordinary experience, is clearly stated in Grassmann's Ausdehnungddtre
von 1844 ; and from a point of view other than that of a Positional Manifold it has been
carefully elaborated by Veronese {loc, cit,),
Homersham Cox constructs a linear algebra [cf. § 22] analogous to Clifford's
Biquatemionsy which applies to Hyperbolic Geometry of two and three and higher
dimensions. He also points out the applicability of Grassmann's Inner Multiplication
for the expression of the distance formuke both in ^Elliptic and Hyperbolic Space ; and
applies it to the metrical theory of systems of forces. His whole paper is most suggestive
[cf. notes, p. 346 and at the end of this volume].
Buchheim states the distance formulea for both Elliptic and Hyperbolic Space in the
same form as they are given in this chapter, with unimportant variations in notation. He
then deduces Clifford's theory of parallel lines ; and proceeds to investigate the theory of
screws in Elliptic and Hyperbolic Space of three dimensions. In his last paper he obtains
an important theorem respecting the motion of a rigid body in Elliptic Space of 2/i— 1
dimensions. Many of his results are deduced by the aid of Biquatemions, and of Cayley's
Algebra of Matrices. A further account of his important papers is given in the note at the
end of the voluma
1
CHAPTER II.
Elliptic Geometry.
214. Introductory. In the following application of the formula of
the Calculus of Extension to the investigation of Elliptic Qeometry the
polar form will be exclusively considered. Most of the theorems and investi-
gations apply, mutatis midandts, to both forms. But each form requires its
own special explanations, which though important geometrically are only
remotely possessed of any algebraic interest. So to avoid prolixity one form
is adhered to.
The space spoken of throughout this chapter will be of i/ — 1 dimensions
where v is any number. It is the merit of this Calculus that the general
formulae for p—1 dimensions are as simple and short as those for two or for
three dimensions.
215. Triangles. (1) Let the terms a, b, c denote three points; there
are eight modes of associating the pairs of intercepts [c£ § 206 (8)] joining
each pair of points ; namely, using lengths as named, that by associating
be, ca, ab; or W7 — be, 7r7 — ca, Try — 06 ; or tp/ — 6c, ca, ab; or be, iry— ca,
TPy — ofc ; and so on.
(2) Let the angle a between the two intercepts ab and ac be defined to
be that angle (out of the two supplementary alternatives) given by
[cf. § 211 (6)]
"" \/{(a6 I aft) (oc ( ac)} '
Similarly for the angles /S and y.
Thus the angle between ab and Try — ca is found by putting — c for c in
the above and is
-(ailoc)
ofm""* i ! — 1
^{(ab\ab)(ac\ac)]*
that is Tr'-o.
Let the angles a, /9,yhe associated with the intercepts 6c, ca, ab ; and
let this system of intercepts and angles be called the triangle a6c.
24—2
372 ELLIPTIC GEOMETRY. [CHAP. IL
(3) Now (ab \ac) = (a \a) (b \c) - (a |6) (a Ic).
Also [cf. § 206 (8)] _
. ab / (ablab) . ac / (ac\ac) 1
7 V(a|a)(6|6) 7 V(a|a)(c|c)'
, ab (a\b) ac (a\c)
and cos — = -777 -j — r->l~TL\) > ^^^ — ~ "777 — i — \ / I \) •
7 's/{(a\a){b\b)\' jy ^{a\a)(c\c)]
TT 6<3 ab ac ^ . ab . 04)
rlence cos — = cos — cos — h sin — sm — cos a ;
y 7 7 7 7
with similar formulse for yff and y.
(4) When a = 0, then c is coUinear with a and 6. Also (ah\ac) is
positive : hence we can write either c^ ^a + b, or c^ — ^a + b, where f is
positive. In the first case by § 206 (9) c lies in the intercept ab ; in the
second case, since b^^c-^- ^a,b lies in the intercept ac,
. , be ab -^ ac
Also cos — = cos - - — .
_ _ _ 7 'y_ _ _
Thus be =^ ab — ac in the first case, and bc = ac — ab in the second case.
(5) Let a\ b\ c' stand for - a, - 6, — c respectively. Then
_ ^ 7 " V{(6' |6') (c' |c2[~ V{(6 |6)(c Ic)} " ''**^ 7 *
Thus feV = fee. Similarly c'a' = ca, a'V = ofe.
Again it is easy to see from (3) that the angle between aV and a'c' is a;
and so on. Hence the triangle aVc is the same as the triangle abCy both in
its sides and angles and angular points. The two are therefore identical.
(6) Consider the triangle abc, which by subsection (6) is the same as
aVc', Its sides are easily seen to be related to those of abc as follows :
Vc* = be, da = irrf-- ca, ab* —iry — ab.
Hence by subsection (3) its angles are a, tt — >ff, tt — y.
Similarly the triangle afe'c, or a'bc\ has sides Try — 6c, ca, wy — ab, and
angles ir — a, J3,ir — y.
And the triangle abc\ or a'Vc, has sides Try —be, Try— ca, ab, and angles
7r — a, 7r — yff, y.
(7) Hence of the eight possible cases mentioned in subsection (1) only
four can have angles associated with them in accordance with the convention
of subsection (2). Accordingly three points will be said to define four
triangles, where a triangle is taken to mean three determinate intercepts
and three angles between each pair of intercepts. The triangle defined
by the terms a, b,e will be taken to mean the triangle with the intercepts
(fe|c), {e\a), (a|fe) as sides, and will be called the triangle abe. The other
triangles defined by the poirvta a, b, e are the triangles a'bc (or ab'c\
ab'e (or a'bc'), abc' (or a'b'c).
215] TRIANGLES. 873
There are two main cases to be considered : fii-stly when one of the four
triangles defined by the points a, 6, c has all its sides less than ^'rry, that is
to say, has the three lengths D (be), D {ca\ D (ab) for its sides [cf. § 204 (2)] ;
secondly, when one at least of the sides of each of the four triangles is
greater than Jth/.
(8) Case I. Let no one of a, b, c be divided from the other two by their
polar planes, then [cf. § 205 (3)] (6 |c), (c\a), (a\b) may be assumed to be of
the same sign; and this sign must be positive. Hence bc = D (6c), ca = D (ca),
ab = D (ah). Thus one triangle (the triangle abc) is formed by the intercepts
of the lengths D (be), D (ca), D (ab) \ each being less than ^iry.
Then by subsection (6) the other three triangles formed by the three
points are (i) that formed by the intercepts D (6c), iry — D (ca), wy — D (ah),
with angles a, tt — yff , 7r — y ; (ii) that formed by the intercepts iry—D (be),
D(ca), Try -D (aft), with angles tt — a, /3, tt — y; (iii) that formed by the
intercepts iry- D (be), Try — D (ea), D (ab), with angles tt — a, tt — yff, y.
(9) Each of these last three triangles has two sides greater than ^iry.
Let the triangle with each side less than ^iry be called the principal triangle
ahe, let the other three be called the secondary triangles.
(10) Case II. Assume that a is divided from b and c by the polar planes
of 6 and c. Then [cf. § 205 (3)] we may assume (6|c) and (a\b) to be
positive, and (a \ e) negative. Hence be^D (be), ca = Try — i) (ca), ab==D (ab).
Also (a6 1 ae) {—(a\a)(b\e)—(a\b)(a\ e)} is positive ;
(6c|6a) {=(b |fe)(c|a) — (a|6)(fe|c)} is negative;
(ca left) {= (c |c) (a \b) — (6 |c) (c \a)] is positive.
Thus, considering the triangle ahe, the angles a and y are acute, and JS
is obtuse; and the obtuse angle is opposite to the side greater than ^Tny.
The other three triangles, defined by the points a, b, e, are (i) that formed
by D (be), D (ca), iry— D (ah), with angles a, tr — J3, ir — y. This triangle
has one side, namely Try — /) (ab), greater than ^iry, and one obtuse angle,
TT — y, opposite to it. (ii) The triangle formed by Try — D (be), D {ea),
D(ah), with angles tt — a, tt — yff, y. This triangle has one side, namely
Try — D (be), greater than ^Try, and one obtuse angle, namely Tr — a, opposite
to it. (iii) The triangle formed by Try — D (6c), Try — D(ca), Try — D(a6),
with the angles tt — a, yff , tt — y. This triangle has all its sides greater than
^Try, and all its angles obtuse.
(11) Thus in this case the points a, 6, c define three triangles each with
one side greater than ^Try, and one triangle with all its sides greater than ^Try.
Call this case, the case with no principal triangle. This possibility respecting
triangles in elliptic space of the polar form has apparently been overlooked.
Let the set, of three triangles, each with one side greater than ^Try, be
called the principal set.
374
ELLIPTIC GEOMETRY.
[chap. IL
216. Further Fobmulji: for Triangles. (1) The fcwo typical trans-
formations, from which the further formulae connecting the sides and angles
are deduced, are
(a,a)(a6c|a6c) = (a6|a6)(ac|ac) — (a6|ac)* (i);
and (6 \c) (abo\abc) = {be \ba) {ca \cb) + {ah \ac){bc \bc) (ii).
Both of these formulae can be proved by mere multiplication. Thus for
instance [c£ § 120]
{bc\ba) {ca cb) + {ah |ac) (6c |6c)
= {(6 16) (c \a) - (a \b) (6 \o)] {{c \c) {a \b) - (6 |c) (c \a)]
+ {{a\a){b\c)^{a\b){o\a)][{b\b){c\c)--{b\cy]
= (6'c) {2(6, c)(c a)(al6)+(a»(6|6)(c|c)-(a|a)(6|c)«-(6|6)(c|a)«
«(c|c)(a|6)-}
= (6|c)(a6c |a6c).
/ox o- /fi 5 ) VK^ft |a6) (ac lac) -(a6i ac)»>
(2) Since sm « = V{1 - cos' a] = ^^ -^^l^i^) ^ l^' .
it follows from equation (i) of subsection (1) that
sina= /(«I«)(^^?L^^)
V (a6|a6)(ac(ac)*
6c __ /J6c|6c)
"V (61
But
sm
Hence
7 -v (6|6)(C|C)'
sin a _ sin yff _ sin y
. 6c . ca . a6
sin — sm — sm —
7 7 7
_ /{a \a) (6 [6) (c \c) {abc \abc)
"" V (6c 1 6c) {ca \ ca) {ab \ ab)
(3) From equation (ii) of subsection (1)
6c
that is,
sin fi sin y cos — = cos fi cos y + cos a ;
6^
cos a = — cos jS cos y + sin yff sin y cos — ,
with two similar equations.
(4) If a, 6, c be at unit intensity then [cf. § 120 (1) and § 210 (2)]
ac I
(a6c|a6c) =
1,
a6
cos —
7
a6 -
cos — , 1,
7
cuy be
cos — , cos —
7 7
cos —
7
6^
cos —
7
This determinant is the sqiiare of the well known function, which in
Spherical Trigonometry is sometimes called the Staudtian of the triangle.
216, 217] FURTHER FORMULiE FOR TRIANGLES. 375
(5) It is evident that the usual formulse of Spherical Trigonometry,
for example Napier's Analogies, hold for triangles in Elliptic Geometry. For
these formulae are mere algebraic deductions from the fundamental formulas
of § 215 (3) and of subsections (2) and (3) of this article.
(6) Let a circle be defined to be a curve line [cf. § 67 (4)] in a two-
dimensional subregion, such that each point of it is at the same distance (its
radius) from a point (its centre) in the subregion. Then it follows from
subsection (2) that the perimeter of a circle of radius p is 2*0^ sin ^ .
y
For consider the chord pq, subtending an angle a at the centre. Draw
cl perpendicular to pq.
Then, since by symmetry I is the middle point otpq,
. PQ .pi . a . cp . a . p
sin s^ = sm-s— = sm TT sm -^ = sin ^ sin - .
zy 7 Z y 27
Therefore when a is made small enough,
^ = a7sin^.
Accordingly, assuming that the length of the arc of a curve is to be
reckoned as ultimately equal to the chord joining its extremities, the
circumference of the circle = ^pg, ultimately, = 7 sin - 2a = 27r7 sin - .
217. Points inside a Triangle. (1) Consider the triangle abc, that
is, the triangle with its sides formed by the intercepts (6|C), (c|a), (aj6).
Any point of the form Xa + fib + vc, where X, fi, v are of the same sign, will
be said to be inside the triangle. Other points of this form will be said to
be outside the triangle.
(2) To prove that any straight line, in the two dimensional subregion
defined by a, 6 and c, cuts the sides of the triangle, either two internally
and one externally, or all three externally.
Write |) = \a + /a6 + i/c ; and let px be any line through p and another
point X in the two dimensional region. Without loss of generality we may
consider that the complete manifold [cf. § 103 (3)] is the two-dimensional
region defined by a, 6, c.
Then pa? . 6c = Xcw? . be + fAx . be 4- vex . be
= {X (xea) — [i {ai)e)] 6 4- {X {xab) — v {xbe)] e ;
px,ea — {/A {xab) — v {xea)] c + {/* (ic6c) — X {xea)] a ;
px.ab = {v {xbe) — X {xab)] a + {i^ {xea) — fi {xah)] 6.
Let ^1 = A* {xab) — v {xea), 02 = v {xbe) — X {xah),
Ot — X {xea) — /* {xbe).
Hence px .bc = fij> — 0^, px ,ea'=i O^e — 0^, px ,ab = 0^ — 0fi.
376 ELLIPTIC GEOMETRY. [CHAP. II.
Now px , be is the point of intersection of px and be ; and if 0^ and 0^ are
of the same sign, this point is external to the intercept (b\c)] and if 0^ and
0^ are of opposite sign, the point is vdthin the intercept (6 |c). But 01,0^^ 0^
are either all three of the same sign, or two are of one sign and the third of
the opposite sign. Hence the proposition is evident.
(3) Any line in the two dimensional region, which contains a point
inside the triangle, cuts two of the Bides internally and one externally ; also
conversely. With the notation of the previous subsection, assume that p
lies within the triangle. Then \, /*, v may be assumed to be all positive.
Also without any loss of generality, x may be assumed to be on the line be,
so that (xbc) = 0.
Then 0i = fi(xab)-'v(xca), 02 = — \{xab), 0^ = \{xca). Hence, if {xab)
and {xca) are of the same sign, 0^ and 0^ are of opposite signs ; also, if {ocab)
and {xca) are of opposite signs, 0^ is of opposite sign to both 0^ and &,.
Hence in either case the first part of the proposition is true.
To prove the converse, assume that the sides (c|a) and (a|6) are cut
internally at the points cut + 7c, a^a 4- fij) ; where a, 7, «!, fix can be assumed
to be all positive. Then any point on the straight line can be written in the
form f (aa + 7c) + 17 (a^a + ;9i6). Hence all points, for which f and 77 are of
the same sign, lie within the triangle.
218. Oval Quadrics. (1) If three points a, 6, c, lie within [c£
§ 82 (1)] a closed quadric, (a$a?)' = 0, then the quadric cuts all of the sides
of one of the triangles defined by the points a, 6, c externally.
For [cf. § 208 (2)] we may assume (a$6$c), (a$c$a), (a][a][6) to be all
positive, when (a j[d?)' is positive, x being a point within the quadric. Now
with this assumption as to the terms a, 6, 0, consider the triangle iJihc. Let
any side be cut the quadric in a point fib + ve. Then
M' {^V>f + 2/41/ (a$6][c) + I/* (a$c)' = 0.
Thus the two roots for fiw given by this equation are both negative.
Hence any side (6 \e) of the triangle cd>e is cut by the quadric in two external
points. It follows that the sides of any of the remaining three triangles
defined by the points a, 6, c are cut two internally and one externally.
(2) An oval* quadric is a quadric which cuts externally the sides of any
principal [cf. §215 (9)] triangle (jbe, of which the three angular points lie
within it.
(3) Let a sphere be defined to be a surface locus contained in the
complete manifold [§ 67 (1)], such that every point of it lies at a given
distance (the radius) from a given point (the centre).
* Oval quadrics have not, as far as I am aware, been previously defined. In the special
case of Euclidean space of three dimensions, ellipsoids and hyperboloids of two sheets are both
closed quadrics ; but only ellipsoids are oval quadrics.
/
f
218] OVAL QUADRICS. 377
A sphere is a closed quadric. For if e be the centre and p the radius,
the equation of the sphere is
.-, [/ , V = cos' - , that is, (e \xf — (e\e) (x x) cos* - = 0.
{x\x){e\e) 7 ^ ^ ^ ^ /\ / y
Now if y be a point at a distance from e less than p, then
. I |y . V > cos' - ; hence (e ly)* — (y \y) (e \e) cos' - is positive.
Also there must be two real points on any line through y which lie on the
surface. For, let any line through y cut the plane •€ in e, so that (e |e') = 0.
Then any point z on this line can be written y + ^e. Hence
(e'^)'-(^i^)(e|e)cos'^ = |(c|y)'-(yIy)(e|e)cos'^
Hence, since -
- 2^(y\e'){e |e)co8' ^ - f'(c' \e') {e |(?)cos»^ .
7 7
{^\yy^{y\y)(fi\^)^^ \ is positive, it is alwajrs possible to
7j
find two real values of f for which {e \zf — {ziz) (e \e) cos' - = 0. Accordingly
7
[c£ § 82 (1)] any point at a distance from the centre less than the radius
is within the sphere, and for such points (e |y)' — (y |y) (e \e) cos' - is positive.
7
(4) A sphere of radius less than Jth/ is an oval quadric. For let e be
the centre of the sphere, and x and y two points within it. Then by (3)
the two intercepts D (ex), and D {ey) both lie within the sphere and are cut
externally by it. Now let the intercepts (e\x) and (e\y) be these intercepts,
so that (e\x) and (e\y) are both positive. Then the triangle exy has two
sides cut externally by the sphere, and hence by (1) the third side (x \y) is
cut externally.
Ti^ ary ex ey , . ex . ey ^
But cos -^ = cos — cos — + sm — sm — cos 0,
7 7 7 7 7
where 0 is the angle at e of the triangle exy.
,T xu ex ey . ex , ey
Hence cos -^ > cos ~- cos ~ — sin — sm -^
7 7 7 7 7
ex-\-ei/
> cos - "^ .
7
Hence xyKex-^-eyK ^iry ; since e^ and ey are by hypothesis each less
than J-TTy.
Thus xy^s^D (jcy). Hence that intercept joining any two points within
the sphere, which is cut externally by the sphere, is the shortest intercept.
Hence the sphere is an oval quadric.
378 ELLIPTIC GEOMETRY. [CHAP. II.
(5) It is also evident by the proof of the preceding subsection that any
sphere of radius greater than ^iry is not an oval quadric. Hence also it is
easy to prove that any oval quadric can be completely contained within
some sphere of radius ^iry,
(6) Furthermore it follows from (1) and (4) that any three points lying
within a sphere of radius ^tt/ define a principal triangle.
219. Further Properties of Triangles. (1) Two angles of a
principal triangle [cf. § 215 (9)] cannot be obtuse. For if possible let a
and JS be both obtuse. Then from § 215 (3)
be ca ah , ab , ca
cos — = cos — cos -^ + sm — sin — cos a,
ca be ab , . be . ab r%
cos — = cos — cos — h sin — sm — cos a.
ty y *y y y
TT 1. i.L ^ ca db . ca be ab ..
Hence both cos cos — cos — and cos cos — cos — are negative,
y 77 'y_'^'^_
since cos a and cos JS are negative. But cos — and cos — are both
be
positive by hypothesis and one of them must be the greater, say cos — .
Then cos cos - cos — has the sign of cos—, and is therefore positive. 1
y y y . y . , , !
Hence there cannot be two obtuse angles in a principal triangle. It has
been proved [cf. § 215 (11)] that, if no principal triangle exist, the triangles
of the piincipal set defined by a, 6, c have each only one obtuse angle,
while the remaining triangle has three obtuse angles.
(2) In any triangle abe [cf. § 216 (7)] if j3 and y be both acute or both
obtuse, the foot of the perpendicular from a on to 6c falls within the in-
tercept (6 |c) ; otherwise it falls without the intercept (6 |c).
For let p = X6 + /ic be the foot of this peipendicular. Then
(ap\bc) = \(ab \be) -{- fi(ac \be) = 0.
Hence we may write p = (ca \cb)b + (ba \be) c.
Now JS and y are respectively acute or obtuse according as (ba \bc) and •
(ca \cb) are positive or negative [cf. § 215 (2)]. Hence the proposition.
(3) If the angles jS and y be both acute, the triangles abp and a/y) have
jS and y respectively as angles (and not ir — jS and ir — y), also the sum of |
their angles at a is equal to a. This proposition is easily seen to be true.
(4) The sum of the three angles of any principal triangle, or of a triangle
from a principal set is greater than two right-angles.
Firstly, let the angle y be a right-angle, and let a and jS be acute.
219, 220] FURTHER PROPERTIES OF TRIANGLES. 379
Then by one of Napier's Analogies,
. bc^ ca ^ be ^ ca
cos J cos J
tan i(a + j9)=^ _ ^ cot Jy = m^'^ ^ .
. bc + ca , 6c4-ca
cos * — cos i
Now since be and ca are each less than ^tt/, it follows that
.bc^ ca /bc + ca
cos 4 > cos ^ .
7 7
Hence a-^- jS >^. Hence a + yff + y > tt.
Secondly, let abc be any principal triangle or a triangle from a principal
set. Then at least two of its angles are acute, say a and fi. Draw a
perpendicular cd from c on to the opposite side. Then d lies between
a and 6 on the intercept (a|6), and ahc is divided into two right-angled
triangles. Hence obviously from subsection (3) the theorem holds for the
triangle ahc.
220. Planes one-sided. (I) It has been proved in § 203 (1) that a
plane does not divide space. An investigation of the meaning to be attached
to the idea of the sides of a plane is therefore required.
Let two points a and b be said to be on the same side of a plane P, when
the intercept D (ah) does not contain the point of intersection of ah and P,
that is to say, the point ah . P.
Conversely when the intercept D {ah) does contain the point ab . P, let
a and b be said to be on opposite sides of the plane.
(2) Suppose that a and ( are on opposite sides of the plane, but that
they each approach the plane along the line ah so that D(ah) diminishes
and ultimately vanishes. Then in the limit a and 6, though coincident in
position, both lie on the plane on opposite sides of it.
Thus a plane can be considered to have two sides in the sense, that at
each point of the plane there may be considered to be two coincident points
on opposite sides of the plane. This idea can obviously be extended to any
surface.
(3) If a be any point on a plane P, then a and — a may be considered
as symbolizing the two coincident points on opposite sides of the plane.
For let b be any other point not on the plane ; and assume, for example,
that (a 1 6) is positive. Write a' for —a. Then if a be considered to be on
the same side of the plane as b, the intercept (a |&) does not contain a' (by
the definition of subsection (1)), the intercept (a' \b) does not contain a (since
a'b is the length of the long (polar) intercept between a' and 6, namely.
Try — D (ah)), and the straight line is completed by the indefinitely small
intercept D (aa') which passes through the plane.
880 ELLIPTIC GEOMETRY. [CHAP. II.
Thus if b be a given representation of the point b, which is taken as the
standard representation, a is on the same side as the point b of any plane on
which a lies when {a\b) is positive, and is on the opposite side when (a |6) is
negative.
It must be carefully noticed that the choice of sides for a and —a
depends not only on the position of the point 6, but also on the special term b
which represents the point. For - b represents the same point, and if — &
be taken as the standard representation, a and — a would according to the
above definition change sides of any plane on which they lie.
(4) Suppose that a sphere of radius ^iry be described cutting the plane,
and that attention be confined to points within this sphere. Then [cf. § 218
(6)] any three such points, a, b, c, define a principal triangle : let it be the
triangle abc.
Now if a and b be on the same side of the plane, then c is on the same
side of the plane as a or on the opposite side of it, according as c is on the
same side as 6 or on the opposite side.
For the plane cuts the two dimensional region dbc in a straight line,
and by hypothesis this straight line cuts the intercept (a\b) externally, hence
by § 217 (2) and (3) it cuts the other two intercepts, (ajc), (6 |c), both
externally or both internally.
Thus when attention is confined to the space within this sphere, the
ordinary ideas concerning the two sides of a plane hold good.
(5) But if the points a, b, c do not define a principal triangle, let the
triangle abc be one of the principal set. Assume that (a {6), (a|c) are
positive and that (6 c) is negative. Now the straight line, in which any
plane cuts the region abc, must cut the sides of the triangle abc either
all externally or two internally and one externally.
If the line cut all the sides externally, it cuts D (ab), D (ac) externally
and D(bc) internally. Hence a and 6 are on the same side of the plane,
also a and c ; but 6 and c are on opposite sides.
If the line cut (a !6), (a \c) internally and (6 \c) externally, it cuts D (oft),
D (ac), D (be) all internally. Hence any two of the three are on opposite
sides of the plane to each other.
If the line cut (a |6), (b c) internally and (a \c) externally, it cuts D (ab)
internally, and D (be), D (ac) externally. Hence c and b ai-e on the same
side of the plane, also c and a ; but a and 6 are on opposite sides. Similarly
if the line cut (a c), (b \c) internally and (a b) externally, then c and 6 are
on the same side of the plane, also b and a ; but a and c are on opposite
sides.
Hence, when three points do not form a principal triangle, the ordinary
ideas concerning a plane dividing space cannot apply.
220] PLANES ONE-SIDED. 881
(6) It has been defined in (3) that if the point a lie on the plane P and
h be another point not on the plane, then the term a symbolizes a point on
the same side as the point 6, when (a |6) is positive.
Let c be another point on the plane so that the triangle aihc is a principal
triangle. Then by hypothesis (6|c) is positive, and the teim c symbolizes
a point on P on the same side as the point 6. Hence, assuming that the
theorems of subsections (4) and (5) are to hold when two angular points
are on the plane, a and c are on the same side of the plane when (a|c) is
positive.
(7) If a and c be two points on the plane P and on the same side of it,
then the point \a +/ac is defined to describe a straight line without cutting the
plane, when any two neighbouring points of the line successively produced
by the gradual variation of \ and /Lt are on the same side of the plane.
Suppose that X varies from 1 to 0 as /t varies from 0 to 1, then the
intercept (a|c) is described without cutting the plane. Also every point on
this intercept is on the same side as both a and c. But now starting from
c let the moving point describe the other intercept without cutting the
plane. Then \ must vary from 0 to — 1 while /Lt varies from 1 to 0. But
the final point reached is — a. Thus a moving point, starting from a and
traversing a complete straight line drawn on the plane without cutting the
plane, ends at — a, that is on the opposite side of the plane.
Again, if Q be another plane cutting P, and the subplane of inter-
section does not cut either P or Q, then when the moving point starting
from a has made a complete circuit of a straight line lying in the subplane
PQ it is on the opposite side both of P and Q to a.
In order to understand the full nieaning of this property, consider for
example space of three dimensions. Let the two sides of P at a be
called the upper and under side, and the two sides of Q at a be called
the right and left side. Let a dial with a pointer lie in the plane P at a
with face upwards and pointer pointing to the right. Let the dial be carried
round the straight line of intersection of the planes so that in neighbour-
ing positions both face and pointer respectively look to the same sides of
the two planes. Then, when the complete circuit has been made, the dial at
a is face downwards and the pointer points to the left.
The property of planes proved in this subsection is expressed by saying
that planes are one-sided. The discovery of this property of planes in the
polar form of elliptic geometry is due to Klein*.
(8) The definition in subsection (7) of a straight line drawn on a plane
without cutting it can obviously be applied to any curve-line drawn on the
plane. Also by the method of (7) it is easy to prove that a point, starting from
* Of. Math, AnnaL, Vol. VI.
882 ELLIPTIC OEOMETRT. [CHAP. IL
a aud describing a closed curve on a plane P, returns to a or to — a according
as the closed curve cuts the subplane of intersection of P and the polar plane
of a (that is, the subplane P |a) an even or an odd number of times.
221. Angles between Planes. (1) Since in Elliptic Geometry the
absolute is imaginary, the separation [cf. § 211 (2)] between planes must
necessarily be measured by the trigonometrical formula and not by the
hyperbolic formula. The same applies to the separation between any two
subregions, when the idea of a measure of separation between them can be
applied [cf. § 211 (6)]. Let the measure of the separation between planes or
between subregions (excluding points) be called the angle between them.
Thus the angle between two planes X and Y is one of the two supplementary
angles (less than tt).
±(X\Y)
*^ V{(^|Z)(F|7)r
(2) Let < XY stand for that one of the two supplementary angles
between X and Y which is defined by
cos<ZF=- -<^l-^ -
cos<A/ ^{(X|jr)(F|10}*
(3) The points |X(=a?) and |F(=y) are the absolute poles of the
planes X and F. The length xy of the intercept {x \y) is given by
cos -^ = ,., , Iv , .. = cos < X\ .
7 V{(^N)(y|y)}
Hence ^ = <XF.
7
(4) If ^ be a third plane, the angles between the subplanes XY and XZ
are the two supplementary angles (less than tt) defined by
±(XT\XZ)
COS
— 1
s^{(XY\XY)(XZ\XZ)}'
These angles are the same as those between the lines ay and az, where
z^lZ.
222. Stereometrical Triangles. (1) The angles which the planes
A, B, C make with each other, and also the angles which the subplanes
BCy CA, AB make with each other can be associated together by definition,
so as to form what will be called a stereometrical triangle. Let the stereo-
metrical triangle ABC be the association of the three angles < BC, < CA,
< AB, with the three angles a, JS, y, defined by
(AB\AC)
co6a=s
^{{AB\AB)(AC\AC)}'
with two similar equations for J9 and y.
221 — 223] STEREOMETRIGAL TRIANGLES. 383
(2) Then if a = |il, 6 = |-B, c = 1(7, the triangle ahc is the * polar ' triangle
of the stereometrical triangle ABC. Also the angles of the triangle abc are
respectively equal to a, fi, y\ while the sides of the triangle abc are
respectively equal to 7(< BG), y(< CA\ y(< AB).
(3) Accordingly, corresponding to every formula for a triangle defined
by three points there exists a formula for a stereometrical triangle defined
by three planes. Thus the ordinary formulae of Spherical Trigonometry, in
ordinary three dimensional Euclidean space, are shown to hold for the relations
between three planes of any number of dimensions in Elliptic Geometry.
(4) From § 215 (3) it follows that
cos < BC = cos < (M cos < -4-8 + sin < Oil sin < AB cos a ;
with two similar formulse.
Now if the complete space be three dimensional, the subplanes BC, CA,
AB are three straight lines meeting at a point ; and thus a, J9, y correspond
to the ' sides ' of an ordinary three dimensional spherical triangle, while
< BCf < CA, <AB, correspond to the angles.
Thus according to analogy the above formula ought to be
cos < BC = — cos < CA cos < -4B + sin < CA sin < AB cos a.
This difference of sign is explained by noting that the angles to be
associated with the stereometrical triangle ABC were defined by convention
in subsection (1); and that if the angles of the triangle ABC had been
defined to be tt — < BC, etc., and ir — a, etc., the signs of the formulae
obtained would have agreed, when the complete region is of three dimen-
sions, with those of ordinary Spherical Trigonometry.
223. Perpendiculars. (1) Any two mutually normal [cf. § 108 (5)]
points X, y are at the same distance from each other. For since (ar|y) = 0,
cos — = 0, and therefore xy = ^^7. Such points may also be called quad-
rantal. The condition that two lines 06 and ac should be at right-'angles
(or perpendicular) is {ah |ac) = 0.
Lines, or other subregions, which are perpendicular must be carefully
distinguished from lines, or other subregions, which are mutually normal
[cf. § 113 (1)].
(2) Let any region Zp of p — 1 dimensions be cut by a straight line ah
in the point a ; then, if (p— 1) independent lines drawn through a in the
region L^ be perpendicular to a6, all lines drawn through a in the region Zp
are perpendicular to ah. For let op,, ap^^.,. opp-i be the (/) — 1) independent
lines.
Then by hypothesis {ah .op,) = 0 = (ah ap^) = etc.
384 ELLIPTIC QEOMETRT. [CHAP. II.
But Xa + S/fp represents any point in Xp. Hence any line through
a is (/Aiopi + /igopj-f ...).
And {ab\(fiiapi + fi^pi +,..)] = fii{ab\api)'\' fJi^iablap^) -{- ... = 0;
which is the required condition of perpendicularity.
Then ah will be said to be perpendicular to the region Lp, or at right-
angles to it.
(3) Any line perpendicular to the region L^ intersects the supple-
mentary (or complete normal) region |Zp; and conversely, any line inter-
secting both Lf^ and |Zp is perpendicular to both.
For, with the notation of the previous subsection, let ab be the line ; and
let b be the point on the line ab normal to a [cf. § 113 (5)], then 6 is normal
to every point on L^. For, ifp be such a point, (ab\ap) = 0.
Hence (a | a) (b \p) — (a \p) (a 1 6) = 0. Hence (a | a) (6 \p) = 0.
But (a I a) is not in general zero. Hence we must have (6 |p) = 0.
Hence 6 lies in |Zp ; and therefore ab intersects |Zp.
(4) If Pp and P^ be two regions noimal to each other [cf. § 113], and if
a be any point in P^y then any line drawn through a in the region P^ is
perpendicular to the region aP,.
For let a' be any other point in Pp, and b be any point in P^, then by
hypothesis, (a , 6) = (a' \b) = 0.
Hence (aa' \ ab) = (a | a) (a' 1 6) - (a 1 6) (a' | a) = 0.
(5) Let two planes L and M intei*sect in the subplane LM, and Oi be
any point in LM, From aj draw Oil in the plane L perpendicular to the
subplane LM, and draw aim in the plane M perpendicular to the subplane
LM, then the angle between L and M is equal to that between Oi^ and a^m.
For in the subplane LM, which is of i; — 3 dimensions (the space through-
out this chapter being of i^ — 1 dimensions), we can find [cf. § 113 (5)] v — S
other points a,, a^, ... a^^, so that ai, a^, ... a^-g are mutually normal. Also
take I in the line ail to be the point normal to Oi. Then by subsection (3)
I is normal to Oi, Os, ... a„_a; and therefore to every point in LM,
Similarly in the line Oim let m be normal to every point in LM.
Then (oj \a^ = (ai 1^8) = (ctp |aa) = ... = (a^ |a„_,) = 0,
and (ai|f) = (a2 0=... = (a,^,|0 = 0,
and (a^\'m) = (a^\m)= ... =(a,^2|^) = 0.
Also we may write L = (aiOg . . . a^^J), and M = (oiOa . . . a„_,m).
Then from §120(1)
(L |i) = (a, |ai) (oa |a,) ... (I \l), (M Jf ) = (a, |a,) (a, |a,) ... (m Im),
(Z I Jlf ) = (ai |ai) (oa loa) . . . (^ 1^).
224] PERPENDICULARS. 386
Hence if d be the angle between L and M, and ^ between aj, and dim,
co8<?= (X|Jlf) ._ {l\m) _ {a,l\a,m)
>J\(L\L){M\M)]~ ^/{{l\l)(m\m)} ^/{{aj,\a,l){a,m Km)} ~*^'*-
Thus 9 = ^.
T/m.
Corollary, It is also obvious that 5= ^ = — .
(6) Any line perpendicular to any plane L also passes through its
absolute pole [of. subsection (3)].
Thus if any plane M include one perpendicular to L, then from any
point of the subplane LM a perpendicular to L can be drawn lying in M.
For, if M includes one perpendicular to £, it includes the pole \L. Then
any line joining any point in LM to \L must be perpendicular to L and must
lie in M.
Also since \L lies in M, then \M lies in X. Hence this property is
reciprocal. Such planes will be said to be at right angles.
It is obvious that, if two planes are at right angles, their poles are
quadrantal.
(7) If two planes L and L' be each cut perpendicularly by a third plane
M, it follows at once from the formulae for stereometrical triangles investigated
in § 222, that the angle between the subplanes LM and L'M is equal to that
between the planes L and L\
23A. Shortest Distances from Points to Planes. (1) The shortest
distance from a point to a plane is the shortest intercept of the straight line
through the point perpendiculsur to the plane.
For let X be the point, p the foot of the perpendicular, and q any other
point on the plane. Let the terms x and p be so chosen [cf. § 206 (9)] that
osp = D {xp) ; so that xp is the shorter of the two intercepts between x and p.
Then by § 215 (3), cos^ = cos— cos^ . Hence, if p5 be greater than
|7r7, xq is also greater than \iry. Thus the points x, p, q must define a
principal triangle. Let the terms x, p, g be so chosen that xpq is this principal
triangle. Then from the above formula, D (xq) > D (xp).
This length of the perpendicular will be called simply the distance of the
point 6t>m the plane.
(2) It is obvious that the other intercept of the straight line xp is the
longest intercept of a straight line drawn from x to the plane.
(3) The pole of the plane is easily seen to be the point which is frirther
from the plane than any other point, namely at a distance ^iny.
w. 25
386 ELUPnC GEOMETRT. [CHAP. II.
(4) Let p be the distance of the point x from the plane L. Then
\irf — pia the distance between x and the point | L.
Hence sing^cosT^'^U + M^)_ = + (?^)
Hence sin ^ co8(^ ^ j-± {(^|^)(|£||£)}*- ± {(^|^)(X|i)ji>
where, as in the other cases, the ambiguity in sign is to be so determined as
to make sin - positive. With this understanding we may write
. p _ {xL)
^'''r^-[{x\x){L\L)\y
225. Common Perpendicular of Planes. (1) The line joining the
poles \L and \L' of any two planes L and L' is obviously [cf. § 223 (3)]
perpendicular to both planes L and L\ Further, any point on the line L\L'
is normal to any point on the subplane LU. Let the line \L\L' intersect the
planes in I and V. Let a be any point on the subplane LL\ Then it is
easy to prove that al and aV are each perpendicular to the subplane LL\
Hence the angle between al and aV is equal to the angle X between the
planes. Accordingly in the triangle lal\ the two angles at I and V are right-
angles, al and aV are each ^^ry, and a is X. Hence IV = X7.
Fio. 8.
It is to be noted that there are two lengths X/y and (w — X) 7 for D {It) ;
the shortest of the two is meant according to the usual convention.
(2) It is easy to see that D{Uf) is greater than the distance -of any point
X in either plane from the other plane. For let a? lie in X, and draw xp
perpendicular to L\ Then xp passes through \L'. Also the distance from
|Z' to p equals that from \L' to l\ both being \iry ; but that from jX' to 2 is
less than that from |Z' to a?, since the line from {X^ to Hs perpendicular to i.
Hence D{IV) is greater than D(xp),
226] DISTANCES FROM POINTS TO SUBREGIONS. 387
226. Distances from Points to Subreqions*. (1) The least distance
of a point a from a line be can be found. For let p be the foot of the
perpendicular from a to be, and let b be any other point on 6c. Then, by
the same proof as in § 224 (1), the three points a, b, p define a principal
triangle. Let this triangle be abp. Then, as in § 224 (1), D{ap)< D{ab)\
and hence D (ap) is the least distance which it is required to express.
But sin
{aF\aF)
_ ^ , pa . ab . r%
But sin*-- = sm — sin a,
7 7
where JS is the angle at 6 in the triangle ahp, that is, the angle at 6 in the
triangle abc, if the term c be properly chosen.
^_ / (ah\ab)
y''\/(a\a)(b\by
, . ^ /[(b\b)(ahc\abc)}
and '''' ^^ ^/ {(^Ic^Ub^bcr
Hence sm-= ^ ^^\^>^^J^\^y
Therefore, if i^ be the force 6c, the distance (S) of a from the line of F
is riven by
^ . i / {al
7 ^(a.a>
(2) This formula can be extended to give the least distance of any
point a fix)m any subregion Pp of p — 1 dimensions.
For the argument of § 224 (1) still holds, and the least distance is evi-
dently the length of the perpendicular ap from a to the subregion. One, and
only one, such perpendicular line always exists, since [cf. §§ 72 (5) and
223 (3)] it intersects both Pp and | Pp.
Let P be a force on any line through p in the region Pp, and let Pp-.2 be
the subregion in Pp normal to P. Let P^ = PPp-i.
mk • ^ / (oPjaP)
Then 8,n^=^^____.
Now since ap is perpendicular to P^, it passes through the normal point
to Pp in the region aP^,. Let p^ be this point. Then p^ is normal to every
point in Pf^,
* These fonnalflD, and the dedactions from them in snbaeqnent articles, have not been
stated before, as far as I am aware.
26—2
388 ELLIPTIC GEOMETBY, [CHAP. IL
Let o=|>p4-Xp; where the two equations iPf\Pi>) = 0, and {pF) = 0, hold
Therefore ((lP) = (j»^).
Also (oP, I aP,) = (p^P^ \p^P^) = ( p^ \p^) (P^ I P^)
= (aF \aF) (P^\P^), (cf. § 121).
And {P,\P,) = iF\F){P^\P^).
Therefore J^^^W Ja^W
(a\a){P,\P,) {a\a)(F\F)
Hence if £ be the distance of a from the subregion Pp, then
.8 / (al
y V (a \a]
y V (a|a)(Pp|P,)•
(3) This formula includes all the other formulae for distance fix>m a
point. For if P^, denote a point 6, then it becomes
^^y^W {a\a){b\b)'
which is in accordance with § 204 (1).
And if Pp denote the plane L, then since (aL) is numerical,
(aL\aL) = {aLy;
hence the formula becomes
. 8 (aL)
sm - = ^
7 VKa|a)(i|i)}'
in accordance with § 224 (4).
(4) Since ap is perpendicular to Pp, it also intersects |Pp and is perpen-
dicular to it. Let q be this point of intersection. Then D (pq) = iiry, and
D {aq) = ^iry - op.
Thus if h' be the distance from a to |Pp,
cos - = sm - = ^ ;' .^^ .
7 7 V (a|a)(Pp|Pp)
But also by the same formula, replacing P^ by |Pp,
. S'_ /(a|P,|.a|P,)
''°7"V (a;a)(Pp|P,)-
Hence is obtained the formula
{aP,\aP,)H<^\P,\<^\P,)-ip.\a)(P,\P,) (i).
This formula is easily obtained by direct transformation by taking [c£
§113 (7)] p mutually normal points in P^ and v — p mutually normal points
in I Pp as reference points, as in § 229 following.
227. Shortest Distances between Subregions. (1) Let P^ and Q^
be two non-intersecting subregions of the pth and <rth orders respectively, so
that p -h 0- < I'. A line, such that one of its intercepts is a maximum or a
minimum distance between them, is perpendicular to both.
227] SHORTEST DISTANCES BETWEEN SUBREGIONS. 389
For let a, 6, c be three points, and let D {be) be small. Then a, h and c
define a principal triangle : let this triangle be the triangle abc. Then it is
easy to prove firom the formulsB [cf. §§215 and 216] that D(ab) ~ D{ac)
is a small quantity of the second order compared to D (be) when, and only
when, the angle at b is a right angle.
The main proposition follows immediately from this lemma.
(2) Let />><r; then the polar region, |Q,, of Q,, intersects Pp in a
subregion of the (p — <r)th order at least.
Also any line pq, from a point p in this subregion to any point q in Q^,
is perpendicular to Q^, and is of length ^Tpy. Accordingly such perpen-
diculars from Pf, to Q, are of the greatest length possible for the shortest
intercept of perpendiculars from Pp to Q^; but they are not necessarily
perpendicular to Pp.
The polar region, |Pp, of P^ does not in general intersect Q^. Hence
in general there are no such perpendiculars from Q, to P^ of length ^^7.
(3) Let qiy qt, ... 9« be a independent points in Q,. Then any point
^ in Q, can be written 2fg.
Hence [cf. § 226 (2)] the perpendicular B from a? to Pp is given by
,^«_ /{(Sgg)P,|(2fg)P,}
8
Write \ for sin- , and square both sides, and perform the multiplication;
7
then
\^{P, IPp) {f,«(3i l?i)+ ?«•(?. I?.) + - + 2f,f, (q, \q,) + ... }
= fiH?i^P l?i^p) + f." (9.^J9^p) + ••• + 2f A (3,Pp |?^p) + ...}.
If S be a maximum or a minimum for variations of ^ in Q,, then X is a
maximum or minimum for variations of fi, fa, ... f«.
„ d\ ^ d\ d\
Hence ;r«. ~ ^ ~ rr*^ » . . , = ^^ .
3fi 3fi 3f^
Thus by differentiation
{{q,P, |?,Pp) - V (P, 1 P,) (q, \q,)] f, + {(?xP. Ij^p) - V (P, |Pp) {q, \q,)] f,
+ ... = 0;
with (T — 1 other similar equations.
Thus, by eliminating fi, f„ ... f,r, an equation is found for \" of the
form
= 0;
390 ELLIPTIC GEOMETRY. [CHAP. IL
Where a ^MtMl) a ^c^^^^^^lll^P.)
where flu- (p^|p^) , a„-a„- ^p^jp^^ ,
with similar equations defining the other as.
(4) Hence, if P^ and Q^ be two subregions of the pth and <rth orders
respectively (p > a), there are in general a common perpendiculars to the
two subregions, which are the lines of maximum or minimum lengths joining
them.
If Pf, and Q^ had been interchanged in the above reasoning, an equation
of the pth degree (p>a') would have been found. But this equation would
not merely determine the common perpendiculars to Pf, and Q,. For, if S be
the length of the perpendicular from any point in Pp to Q,, then, with the
notation of the previous subsection,
, B d\ S dS
\ = Sm - , ^jr = cos - . -ttt; .
Hence ;rir = 0, when 8 = iiry, as well as when 8 is a maximum or a
Oil
minimum. Thus the infinite number of lines discussed in subsection (2)
fulfil the conditions from which this equation of the pth degree is derived.
(5) A formula can be found which determines the cr feet in Q, of these
perpendiculars. For, if- q^qt, ... q^, be these feet, then in the equation for
V of subsection (3) the cr roots must be [cf. § 226 (2)]
Hence equations must hold of the type {qiPp \ qJPp) = 0, and of the type
(?i I ?«) = 0.
Thus 9i, 92, ... q^ are the one common set of cr polar reciprocal points
[cf. § 66 (6) and § 83 (6)] with respect to the sections by Q^ of the two quadrics
(x \x) = 0, and (xP^ |a?Pp) = 0.
Thus the cr feet in Q^ are mutually normal.
(6) These common perpendiculars all intersect |Pp [cf. § 223 (3)]. These
cr points of intersection with |Pp are also mutually normal.
For any line joining P,, and Q^ must lie in the region Pf^Q^ of the
(p + cr)th order defined by any p + cr reference points, p from Pp and a- from
Q^, Also the common perpendiculars, being perpendicular to Pp, all intersect
the region |Pp (of the (i; — p)th order), and are perpendicular to it. Hence
they all intersect the subregion, Pf,Q^\Pf„ formed by the intersection of |Pp
with PpQ^f, But this subregion is of the crth order. Then the common
perpendiculars of Qa and Pf, are also common perpendiculars of Q^ and
•PpQ(r|Pp, since PpQa\Pp is part of |Pp. But by the previous subsection, if
Pi', p/, ... p/, be the cr feet in PpQ^,\Pf, of these perpendiculars, then
228] SHORTEST DISTANCE BETWEEN SUBREOIONS. 391
i>/> p^i •••i><r' form a mutually normal set of points. Also they are the
one common set of a polar reciprocal points with respect to the sections by
PfQ^ \P^ of the two quadrics {x \x) = 0, and {xQ^ |^*)= 0.
(7) Now, since 2<r< v, the 2cr points gi, g,, ... g^, pi\ pi, ... p/ are
in general independent. Hence the cr lines Piqi,Pi'q%, ..., cut Pp in cr inde-
pendent elements jpi, jpi, ...p^, which define a subregion P^. Then by
subsection (o) pi,pt, ... p<r Are mutually normal. But they are also normal
to pi', pi, . . . pj. Thus [c£ § 113] the cr lines of the cr common perpendiculars
are mutually noimal, so that any point on one is normal to any point on
the other.
(8) The theorems of subsections (5) to (7) can be proved otherwise thus,
assuming subsection (1) and that one common perpendicular exists between
Pft and Q^. For, since this perpendicular (call it F^ intersects P^ at right-
angles, then [cf. § 223 (3)] any line drawn in Pf, through the point of inter-
section intersects the region \Fi, But f> — 1 independent such lines can be
drawn. Thus |Pi intersects Pp in a region of the (/> — l)th order: similarly
it intersects Q, in a region of the {a — l)th order. Let these regions (both
contained in jPi) be called Pp.i and Q^.^. Then by the original assumption
Pp.1 and Q^_i have a common perpendicular. Call it F^. Then P, lies in \Fi
and is therefore normal to it.
Also [Pj intersects P^i and Qa-i in two regions of the (/> — 2)th and
(cr — 2)th order ; and so on. Hence (cr < p) by continuing this process, c
common perpendiculars can be found, all mutually normal.
228. Spheres. (1) Let h be the centre and p the radius of a sphere ;
its equation is
(a?|a:)(6|6)cos*^=(6|a;)".
But [cf. § 110 (4)], (6 1 a?) = 0, is the equation of the polar plane of b with
respect to the absolute.
Hence [cf. § 78 (2)] a sphere is a surface of the second degree, touching
the absolute along the locus of contact of the tangent cone to the absolute
with 6 as vertex.
(2) It is obvious that any point on a sphere of radius p and centre h is
at a distance ^7 — /> from the polar plane of 6, viz. from \h. But \h may be
any plane since 6 may be any point. Hence a sphere of radius p is the locus
of points at constant distances, \iry — p, from a plane.
A plane can be conceived to be the limiting case of a sphere of radius
^Try. For if 6 be the absolute pole of any plane, the equation of the plane is
(x\b) = 0;
and this is the degenerate form of the equation of the sphere, when p is put
equal to ^7.
392 ELLIPTIC GEOMETRY. [CHAP. XL
(3) Every line, perpendicular to a plane and passing through the pole of
the plane with respect to a sphere, passes through the centre of the sphere.
For let b be the centre of the sphere, p its radius and let p be the pole of
the plane with respect to the sphere.
The equation of the plane can be written
Hence [cf. § 110 (4)] the absolute pole of the plane is
p{b\b)coQ*^-b(b\p).
But [cf. § 223 (3)] the perpendicular lines to the plane pass through the
absolute pole. Hence the perpendicular through p to the plane is the line
p\p(b\b)coa*^-b(Tb\p)y,
that is, dropping numerical factors, the line pb. Accordingly this line passes
through b.
Corollary, The perpendicular to a tangent plane of a sphere through its
point of contact passes through the centre of the sphere.
(4) Let the length of a tangent line from any point x to the sphere,
centre 6, radius p, be t. Let the line meet the sphere in p. Then consider-
ing the triangle xpb, by the last proposition, the angle at p is a right-angle.
Hence the triangle xpb may be assumed to be a principal triangle.
Fig. 5.
TT xb xp p r p
Hence cos — = cos -^ cos - = cos — cos - .
ry y y y y
Therefore the lengths of all tangent lines from x to the sphere are equal.
Substituting for cos — , we find
(x \by = co8» - cos» £ (x \x) (b \b).
Hence when t is constant the locus of a? is a sphere concentric with the
original sphere.
228] SPHERES. 393
In order that the tangent r may be real, we must have
— <i.
oo^ ^(x\w){b\b)
Hence / r J/iiLx < cos* - .
(x\w)(b\b) 7
Therefore the point x must be at a distance from b greater than p, that is
to say, must be outside the sphere [c£ § 218 (3)].
(5) The intersection of any plane with a sphere is another sphere of
1/ — 3 dimensions contained in the plane [cf. § 67 (1)].
For let L be the plane, and
(a?|6)»-(a;|a;)(6|6)cos«^ = 0 (i)
be the equation of the sphere.
Then any point p on the perpendicular from b to L [cf. subsection (3)]
can be written b + \\L.
Hence the distance S from p to any point x is given by
8_ (x\py _ {(a?|6) + X(Za?)}«
y^{x\x){p\p) {x\x){{b\b)^-2\i,Lb)^-\\L\L)Y
Now let X lie on the locus of intersection of L with the sphere, then
{xL) = 0, and x satisfies equation (i).
cos'- =
S
(6 \b) cotf £
Hence ^'^ = (^|j,) + 2X(2^^-hl'^Z|Z) (^^>-
Thus the distance of any given point on the line b\L from any point on
the locus of intersection of L and a sphere, centre 6, is constant.
But this must hold for the point L.b\L, where the line b \L intersects
the plane L. Hence the locus of intersection is a sphere of i^ — 3 dimensions,
with the point X . 6 |X as centre.
The radius Si of this sphere (of v — 3 dimensions) is easily proved by
equation (ii) to be given by
^ (fe|6)(i|Z)cos>e
(6) The locus of the intersection of any two spheres is contained in two
planes, the radical planes.
For let {x\x)(b\b)co^^ -(b\xy=:0,
and (x\x)(c\c)co8* — (c\xy^O,
y
be the equations of the two spheres.
894 ELLIPTIC OBOMETRY. [CHAP. tL
Then two planes containing the locus of intersection are given by
(6 \xy (c ic)cos»--(c \xy(b |6)cos»^ = 0 ;
that is, by the two equations,
(6 \x) \/(c |c) cos - ± (c \x) »J(b \h) cos - = 0.
Let these planes be called the radical planes.
(7) These planes are the loci from which equal tangents can be drawn
to the spheres. Also from subsection (5), it follows that the locus of inter-
section of two spheres consists of two spheres of i/ — 3 dimensions.
(8) Spheres cut each other at two angles of intersection, one correspond-
ing to each radical plane.
For consider the radical plane
(6k)V(c|c)cos^-(ck)V(6!6)cose = 0 (iiiX
Then for points on this plane the equations of the two spheres can be
written
(6 \x) = ± V(^]^(6|y) cos ^ .
7
a
(iv),
(c \x) = ± \/(a? \x) (c \c) cos - ,
7
where the same choice (upper or lower sign) determining the ambiguity is to
be made for both equations.
Also the angle o, at which the spheres cut each other along this plsme, is
the angle between the lines hx and ex through a point x on that part of the
locus of intersection contained by the plane.
Tj {xh \xc)
Hence cos o) = ... ,^ '/ — -r .
vK^o|a?6)(a?c \xc)}
Hence, eliminating x by the use of equations (iii) and (iv),
(6|c)-V(6|6)(c|c)cos^cos-
COS l» =
7 7
\/{(6 1 6) (c |c)} sin - sin -
S 9 <r
cos — cos - cos -
-7 7 7
, p , a
sm-sin -
(v),
7 7
where S is the length be of that intercept between the points b and c defined
by Q>\e).
228] SPHERES. 395
Similarly for the other radical plane
(6 \x) V(c I c) cos - + (c la?) ^/(b |6) cos ^ = 0,
the angle to' of intersection between the spheres is given by
(b\c) + V(6 16) (c |c) cos^ cos -
cos 60 =
1 2
V{(6|6)(c|c)}8m^8m^
S p a-
cos - + cos ~ cos -
1 3^^ (vi).
sin «- sm -
7 7
Hence if 8' be the length of the other intercept between 6 and c, so that
8 + 8' = Try, then
cos cos- cos -
cos (it - ob!) = '^ 77
. p . c
sin - sm -
7 7
This equation exhibits the identity of type between the formulae of the
equations (v) and (vi).
Corollary. It may be noted as exemplifying equations (v) and (vi)
that, if « = TT , then cos - = cos^ cos - .
2 7 7 7
Hence cos «' = 2 cot ^ cot - .
7 7
This illustrates the fact that both parts of the intersection are not
necessarily simultaneously real.
(9) Let it be assumed that (6 1 c) is positive, so that 8 = D (6c) < ^Try.
Also by definition, p and a are both less than ^177.
Then the two spheres have real or imaginary intersections oh the
corresponding radical planes, according as cos a> and cos 01/ are numerically
less or greater than unity.
Now cos ©' is positive ; and cos «' < 1,
.- DCbc) , p <T , p . a
II cos — ^^ — - + cos - cos - < sm - sm — ;
7 1 If y "i
,, , . .- D{hc) Try-p — a-
that IS, if cos — ^ — < cos — ^ — ;
7 7
that is, if J)(bc) + p + a'>iry (vii).
Also if cos w be less than unity, cos m is necessarily numerically less than
unity. Hence if w' be real, a> is also real ; and both parts of the intersection
are real.
396 ELLIPTIC GEOMETRY. [CHAP. IL
Thus the condition that the intersections of the spheres with both radical
planes may be real is
D (6c) + p + cr > Try.
(10) If i> (be) + p + <r < iry, then one of the intersections is imaginary.
The condition that the other may be real is
jD (6c) p a-
cos ^^ — - — COS - COS -
_1< — 1 1-^<1.
. p . <r
sm - sm -
7 7
Hence cos — ^^ — ^^ > cos ,
7 7
p '^ a
< COS ' .
7
Therefore the condition that one intersection (at least) may be real is
p/-'cr< D{bc)<p-\-a (viii).
(11) It follows from the inequality (vii) of subsection (9) that two oval
spheres cannot have two real intersections. For D (be) < ^iry, and by § 218 (4)
for oval spheres p and a are both less than ^tt/.
Hence D (be) + p + <r< iry.
(12) Let the sphere, centre c, reduce to the plane L, so that we may
write |c = X, and a- = ^y.
Hence the angle a> at which the plane cuts the sphere, centre 6, is given
by
sm -- cos a = ^ '
7^""- {ib\b)(L\L)}i'
If the plane cut the sphere at right-angles, (bL) = 0. Hence the plane
contains the centre of the sphere.
(13) If the plane touches the sphere, a> = 0. Hence the plane-equation
[cf. § 78 (8)] of the sphere, of centre 6 and of radius p, is
(6|6)(i|i)sin«2 = (6i)».
7
If the sphere be defined as the locus of equal distance a from the plane
JS, then remembering that 5 = |6, and that p + a-^^^y, this equation be-
comes
(i>|i»(i|X)o».?=(S,i,..
Similarly the point equation of the sphere takes the two forms
(a?|fl?)(6|6)cos«^ = (6k)>,
7
and (x \x)(B\ B) sin« - = (xBy.
7
229] PARALLEL SUBREOIONS. 397
229. Parallel Subreqions. (1) Let Pp be a subregional element of
the pth order, then the locus of points x, which are at a given distance S from
the subregion P^,, by § 226 (2) is determined by the equation,
This is the equation of a quadric surface. In the special cases in which
Pp is either a point or a plane, the surface reduces to a sphere.
(2) If Pp be neither a point nor a plane, reed generating regions exist on
this surface. For let 6i, 6^, ... &p be /> mutually normal points in P^, each at unit
normal intensity; and let e^+i, ...e^he v—p mutually normal points in |Pp, each
at unit normal intensity. Then &i, e2, . . . &p, e^+i, ...e^ form a set of v mutually
normal points at unit normal intensity. Let Cie^ ... 6p be written for P^y then
(Pp |Pp) = 1. Also let a be written if6 + 2 ve. Then (x \x) = if» + £ fj\
1 p+i 1 p+i
Also
(«Pp|flrPp)= 2 i7x*(^A«i...ep|eA6i...«p)+ 2 2 fjkViL(fiKei...ep\epfii...ep\
Asp+l A»p+1 |&»p+l
where \ and /i are assumed to be unequal in the double summation. But
(e, \ea) = I, for all values of a ; and (e^ |6r) = 0, for all unequal values of a and
r. Hence {e^^i . .. &p \exei . . . 6p) = 1, and {e^Bi . . . ^p j^^^i • . . &p) = 0.
Thus (fl?Pp|flrPp)= 2 v.
p+i
Hence the equation of the surface takes the form
8in« - ifx"- cos»- 2 V = 0.
7 1 7p+i
If p< ^v, then [cf. § (80) (1) and (5)] noting the formation of conjugate
sets of points from reciprocally polar sets, and remembering Sylvester's
theorem [cf. § 82 (6)], it is evident that reed generating regions defined by
p points exist on the surface, that is, regions of /> — 1 dimensiona
If p > ^v, then real generating regions defined by y — /> points exist on
the surfece, that is, regions of v — p — 1 dimensions.
If p=s^p (p even), then reed generating regions defined by p points exist
on the surface, that is, regions of p — 1 dimensions.
(3) Let these real generating regions be said to be parallel to Pp. Thus
a region parallel to Pp is by definition such that the distances from all
points in it to Pp are equal, and has been proved to be of the type Qp or
Qr-p> according oa poT v — p Ib least.
Also from § 226 (4) a surface of equal distance frt>m Pp is also a surfiEu;e
of equal distance from |Pp. Thus all regions parallel to Pp are also parallel
to |Pp, and conversely.
898 ELLIPTIC OEOMETRY. [CHAP. II.
(4) Let the region Q« be parallel to Pf,) where cr is equal to the least of
the two p and 1/ —p. Let q be any point in Q,, and let qp be the perpen-
dicular from Q^ to Pp. Then qp is also the perpendicular from p to Q,.
For if not, let p^ be this perpendicular. Then D {pq) < D (pq). Also fi-om
g', let q'p' be drawn perpendicular to Pp. Then either p' coincides with p, or
^ (i^V) < •'^ (l^O- Hence in any case D (p'q^) < D (pq)* But, since the
region Q^ is parallel to P^, D (p'^) ^ D (pq). Thus pq must be a common
perpendicular of P^ and Qa-
Thus, if for example p be less than v — p,30 that a^p, then Pp is parallel
to Qp, Thus Pp and Qp are mutually parallel to each other. But the same
proof does not shew that |Pp is parallel to Qp. For, by the preceding
subsection, no region parallel to Qp can be of an order greater than the pth ;
and by h3rpothesiB v — p is greater than p. Also, if 8 be the distance of F^
from Qp, the entire region parallel to Qp at a distance ^ttj — B horn Qp most
be contained in |Pp. Hence a subregion Rp of |Pp of the pth order is
parallel to Qp.
Accordingly a distinction must be drawn between the &ct that one
region is parallel to another region, and the fact that two regions are
mutually parallel. Thus with the above notation, Qp is parallel both to P^
and to |Pp; also {pKv — p) Pp and Qp are mutually parallel ; but jPp and Q^
are not mutually parallel, though Pp' (a subregion of |Pp) and Qp are mutually
parallel. The feet of the perpendiculars from all points in Qp to IPp must lie
in this subregion Pp. This agrees with § 227 (7) : the perpendiculars found
by the method of that article must be all equal.
(5) This theory of parallel regions is an extension* of Clifford's^ theory
of parallel lines in Elliptic Space of three dimensions. Consider a straight
line X in a space of i' — 1 dimensions, (1^ ^ 4). Then the regions parallel to
L are also straight lines, whatever be the dimensions of the space, provided
that they are equal to or greater than 3. If the space be of three dimensions,
then only two parallels to L can be drawn through €my given point x, being
the two generating lines of the quadric surface through x of equal distance
from L. But if the space be of more than three dimensions, an indefinite
number of parallels to L can be drawn through any given point. Also the
tangent plane at a to the surface of equal distance from L which passes
through X cuts this surface (a quadric) in another quadric of one lower
dimension. Hence [c£ § 80 (8) and (12)] this quadric is a conical quadric
formed by the parallels through x. Thus in a region of y— 1 dimensions
the parallels through a; to a line L form a conical quadric of 1^ — 3 dimensions
with X as vertex.
* ffiiherto nnnotioed as far as I am aware.
t CI Preliminary Sketch of BiquatenOom, Proe, of Land. Math. Soe., vol. 4, 1878, reprinted
in bis Collected Papere.
CHAPTER III.
Extensive Manifolds and Elliptic Geometry.
230. Intensities of Forces. (1) In considering the special metrical
properties of extensive manifolds we shall confine ourself to three dimensions.
The only regional elements in this case are planar elements and force&
The intensity of a planar element Z is now taken to be [cf. § 211 (5)] (X \X).
The intensity of a force F has now to be determined.
(2) Let* it be defined that the intensity of the force ocy is some function
of the distance xy multiplied by the product of the intensities of x and y.
Thus assume that the intensity of ay is V{(«^ 1^) (y \y)] 4>{?^ \ where the
function <f>Q^) has now to be determined. Now let x and y be at unit
intensity, and let a be any number, then ay = a? (y + cue). Hence
(a;|a?)=l=(y|y); {(i/ + euc)\(j/ + ax)] = {y\y)+2a(x\y) + cf(x\x'
= l + 2acos^ + a».
7
Accordingly the intensity of ajy = the intensity of a? (y + ax)
= n + a* + 2acos^j <f>[x(y + ax)}.
Therefore, if> (xy) = f 1 + a* + 2a cos —j <f>{x{y + ax)].
But 8in^^y±^> = 7 2-— T-
'y (l + a»+2aco8^)*
Hence ^^ = fHV^}^
xy . x(y + ax)
sm — sm ^^ — -
y y
* This reaBoning is very analogons to some reasoning in Homersham Coz*s paper, of.
loe. dt, p. 870.
'
400
EXTENSIVE MANIFOLDS AND ELUFTIC OEOMETBT. [CHAP. III.
But x(j/ + our) can be made to be any length (< Tpy) by choosing a
suitable value for a.
Hence
sin-^
a constant = 1, say.
Therefore whatever points m and y are, the intensity of a;y is
K^k)(yly)}*8in^.
(3) Hence the intensity of the force F is {F\F)y Thus if the force F
be written PQ, where P and Q are planar elements, the intensity oi F is
{(P IP) (Q I Q)}* sin <PQ.
231. Relations between two forces. Let F and F' be any two
forces. In general there are only two lines intersecting the four lines jP, F*^
\F, \F\ These two lines [cf. § 223 (3)] are two common perpendiculars to F
and F' [cf. § 227 (7) and (8)].
Let one perpendicular intersect F and F' in a and b respectively, and let
the other intersect F and F' in c and d. Let ob = S, and cd = ^. Then one
Fio. 6.
of the two is the shortest distance from one line to the other, and the other
is the longest perpendicular distance. Also since ah intersects F, F\ \F, \F',
\ab intersects the same four lines. Hence cd is the line \ab, and oi is
the line \cd. Thus M = ^iry = ac. Hence (a |c) = 0 = (6 |d) = (a \d) = (6 |cX
(2) To prove that*
(Fr) ' ^ ' ^ __(F\r)___ B s;
{{F\F)(F'\r)}^'^^y^^ 7' {(F\F)(F'\F')\^^'^^y'^ y'
Let F^Xac, F' ^\'hd.
(old)
Then
Hence
8 _ (g |6) .,__
'^'y-{(a|a)(6|6)}*' ^'^ ""[(c!c)(d|d)j*-
8 y_
(a|6)(c|d)
But (i'|^ = X»(ac|ac) = V{(a|a)(c|c)-(a|c)>}=X»(a|a)(c|c).
Similarly, (^' |i") = >''*(6|6)(d|d).
* Gf. Homeraham Cox, loc. cit.
231, 232] RELATIONS BETWEEN TWO FORCES. 401
Also (F\F') = XK'{ac\bd) = X\'ia\b)(o'd).
Therefore cos - cos - = p|>)('y.|^/)n •
(3) Again, from (1) we may write cd = fi\ab.
Hence (cd \ cd) = fi^ (ab \ ah).
Therefore [cf. § 204 (1)]
. if
sm —
— jUc\c){d\d)]^
'^"ITt(«l«)(M6)) ■
sm -
7
Also (oftcd) = /A (oft |a6) = ft (a I a) (6 16) sin'-
.sin?»«fl(.W(6|6)(o|c,(.W)'.
But (FF')=7CK' (acbd).
Hence assuming that the ambiguity of sign is so determined as to make
both sides positive,
(FF') .8.8'
{{F\F) (F' \F')}i ~ ^^ y^^^y'
(4) If the lines F and F' intersect, either 8 or 8' vanishes, say 8" = 0.
Then (^r) = 0, and {(^.i^^I^W^'^'y'
This agrees with § 211 (6).
232. Axes of a System of Forces. (1) A system of forces (S) can in
general [cf. ^ 174 (9) and 175 (14)] be reduced in one way and in one way
only to the form*,
S = aiOa + eloiaj.
Then the lines of the forces aiO^ and e \(h(h ^^^ called the axes of the
system (sometimes, the central axes), and the ratio of their intensities,
namely € (or -J , is called the parameter of the system.
Then (SS) = 2€ {ai<h | <h(h)>
and (S\S)={1 + e'){(h(h\<had.
(2) Let 8 denote the system F + €\F, and /ST the system F' + ff\F\
Also with the notation of § 231 let B and S' be the perpendicular distances
between the lines F and F\ reckoned algebraically as to sign.
* Cf. Homersham Cox, loc. eit., p. 370.
w. 26
402 EXTENSIVE MANIFOLDS AND ELUPTIC GEOMETRY. [CHAP. HI.
Then (SS') = (1 + «;) (FF') + (e + ri)(F\ H
= {(^1^) {F' \F')]i |(1 + ev) sin - sin - + (e + 17) cos - 00s -| ;
and (fif \S) = (€ + 97) (iT') + (1 + ^ri)F\F')
= {(^!^)(F|r)l*|(€ + i;)8m^sin| + (l+€i;)cos^cos|}.
(3) The simultaneous equations (fifS') = 0, (S \B') = 0, in general secure
that the axes of S and S' intersect at right angles. For from (2) unless
either e or ^ be ± 1,
cos - cos - = 0 = sin - sin — .
7 7 7 7
Thus S = 0, V — \mr^\ or vice versa. Hence F and F' intersect at right
angles.
Therefore [cf. § 223 (3)] F intersects \F' as well as F' \ and F' intersects
I F as well as F. Also, since (^^0 = 0. ( | F | i?") = I (^^') = 0- Thus | F and
1^' intersect. Also these various pairs of lines [cf. § 223 (3)] intersect at
right anglea
(4) Every dual group contains one pair of systems, and in general onlj'
one pair, such that their axes intersect at right angles.
Let iSi and B^ define the dual group, and let
Then, (88') = 0, becomes
\fJi^ (8i8,) + (\fJL^ + X^O (8,8^) + X^, (iSaSfa) = 0.
Similarly (jSf |iS') = 0, becomes
\iJ^{8,\8,)+{\^ + \^,)(8A8,) + \fi,(8,i8^) = 0.
Hence eliminating fjLi : /ia, the pair of systems are given by the quadratic
for Xi :Xa,
V K-SA) {8, \8,) - (8 A) (8, 1 8,)} - V 1(8 A) (8^ \8,) - (8 A) (8, \ 8,)]
+ \\, {(8 A) (8, \8,) - (8 A) {8, 1 8,)} = 0.
Let this pair of systems be called the central pair of the dual group, and
let the points at which their axes intersect be called the centres of the
group. There are [cf subsection (3)] four centres to a dual group, forming
a complete normal system of points.
If the group be not parabolic [cf. § 172 (9)], the two director forces Di
and A niay be written for 8i and 8^ in the above equation. The equation
then becomes
(AA) {V (A I A) - V(A I A)} = 0.
232] AXES OF A SYSTEM OF FORCES. 403
Hence (consideriog only real groups) there are always two distinct roots to
this equation. But, if i>i and i), be both self-normal, this equation is an
identity. For this exceptional case, c£ § 235 following.
If the group be parabolic. Let Si be any system, and replace fif, by the
single director force D, Then {DSi) = 0, and the equation for Xi : X, becomes
V(i8f,|i>)-|-XxX«(i)|i)) = a
Thus the central pair of the group are D and (i) \D)8i— (Si \D) D.
(5) To find* the locus of the axes of the systems of a dual group. Let
the four centres be e, e^^ e^, e^, forming a complete normal system of unit
points; also let eei and \eei be the axes of one central system, and ee^ and
je^s of the other.
Let 8i = eei-i-€i\eeiy i8i = c«j + e2|ee2, denote this central pair of systems
of the group ; and let ei and 6, be called the principal parameters of the dual
group. Any other system S' of the group can be written
8' = XySi + XjSa = e (\ei + V2) + \e (X,ei«i + Va^s).
Consider the system
S" = (e + f<?3) 0*1^1 + H^%) + e |(e + ^e^) (fhei + fi^).
It is a system of which a central axis intersects the line ee^ at right
angles [cf. § 223 (3)] in the point 6 + fe, ; also its parameter is 6.
But we may assume 6361 = {ee^y e^ = l«^i, \e^i = ^s, \^s = ^i-
Hence /S" = « {(^ - /Ajfe) Ci + (/ia + fhf e) ^g}
But in general we can make S" and S' identical by putting
/^-€/A2?=^ (1),
/i2 + €/[^f=Xa (2),
€fli - fhS == €i\ (3),
e/i, + /^f = €3X3 (4).
Hence, by elimination, we find
(€»-|-I)(6iX,« + 6aV)-€{(6i«+l)Xi»+(€a«+l)V)=0 (6).
This is. a quadratic to find e; the two roots are reciprocals, namely e
and - , corresponding to the two axes of any system.
€
Again, let any point os on either axis of the system 8' be
* Cf. Homersham Cox, loc. eit,
26—2
404 EXTENSIVE MANIFOLDS AND ELLIPTIC GEOMETRY. [CHAP. III.
Then, assuming for example that a? is on the axis (e + ^e,) (jj^Bi + /j^), by
comparing with the original form of /ST',
Also by elimination between equations (1), (2), (3), (4), (6)
(e. - €>) U^ (P + V) = (1 - ^le.) (f ' + &•) f ?. (7).
This surface is the analogue in Elliptic space of the cylindroid. It is
the locus of all the axes of systems of the dual group. All the central axes
intersect at right angles the lines ee^ and {ees which are called the axes of
the dual group.
(6) Equation (5) of the previous subsection can also be found thus.
Assume that ah is the axis of the system 8'.
Thus S'^e (XiC, + Xgeg) + \e (Se^ei + Xj€^j) = a& + € |a6.
Then {8'S') = 2 (cAi' + e^^){ee^ \ee,) = 26 {ah \ab) ;
and {S' |fif')= {(l + €i») V + (l + €,») V} {ee, \ee,)^{l + ^^){ab\ab).
Thus finally
2e 2 (€iV + €,V)
This is equation (5) of the previous subsection.
238. NoN-AxAL Systems of Forces. (1) If a system of forces, S, be
such that (/Sfig)= ± {S \S\ then [cf. § 174 (8) and § 175 (13)] 8 has not a pair
of axes [cf. § 232 (1)] ; provided that 8 be not self-supplementary [cf. § 235,
following], in which case it has an infinite number of pairs of axes.
Such systems may be called non-axal. It will now be proved that all
non-axal systems are imaginary.
(2) No real hyperbolic dual group can contain a real non-axal system.
For let F and F' be the real director forces of this group, and let the non-
axal system be \F -{-X'F'. Then by subsection (1)
\^{F\F)^-2\\' [{F\r)T{FF')]+\'^{F'\F')^0.
Hence from § 231 (2), if S and h' are the lengths of the two common
perpendiculars to F and F\ this equation becomes
But the roots of this equation are necessarily imaginary. Hence the four
non-axal systems, which belong to any real hyperbolic group, are necessarily
imaginary.
1
233, 234] PARALLEL LINES. 405
(3) But any real system must belong to some real hyperbolic groups.
For [of. § 162 (2)] the conjugate with respect to the system of any real line,
not a null line, is a real line. Now the dual group with these two lines as
director lines is a real hjrperbolic group, and contains the real system.
(4) It therefore follows from (2) and (3) that all non-axal systems are
imaginary.
Hence any real system fif, for which (88) = ± (fif | fif), is self-supplementary.
234 Parallel Lines. (1) An interesting case arises [cf. § 231]
with regard to lines with a special relation discovered by ClifiFord*, and
called by him the parallelism of lines [cf. § 229]. It is to be noted that
the parallel lines of Hyperbolic Space [cf. Ch. iv. of this Book] do not exist
(as real lines) in Elliptic Space, and conversely these parallel lines of Elliptic
Space do not exist in Hyperbolic Space.
In general only two lines intersect the four lines F, F\ \F, \F\ But if
these four lines are generators of a quadric, then an infinite number of
lines — namely, the generators of the opposite system — intersect them.
The two lines F and F* have then the peculiarity that an infinite number
of common perpendiculars can be drawn. F and F' will then be proved
to be mutually parallel according to the definitions of § 229 (3) and (4).
(2) Since the four lines are generators of the same quadric a relation [cf.
§ 175 (4) and (5)] of the formf,
must exist.
Taking its supplement, it must be identical with
\ u» \ u
Hence ~ = r ~ ~ ~ <"> •
/t A. /A X
Accordingly X = ± /a, X' = ± /a'.
Firstly, let X = /a, X' «= /*'. The condition becomes
X(^+|^) + V(^' + |F)=0.
Let the relation of F and F' be called ' right parallelism.*
Secondly, let X = — /a, X' = — /a'. The condition becomes
X(J^-1^) + X'(^'-1^') = 0.
Let the relation of F and F* be called ' left parallelism.'
♦ Cf. loe. ctt., p. 870.
t This fonn of the relation between parallel lines was first given by Bachheim, Proc, London
Math. Society f loe. cit.
406 EXTENSIVE MANIFOLDS AND ELLIPTIC GEOMETRY. [CHAP. HI.
(3) Consider the equation
Multiply it successively by F and F\ then
X (J^ 1^) + V [{FF') + (J^ \F')] = 0,
and X [{FF') •^{F\F')]-\- X' {F' \ F') = 0.
Hence by eliminating X : X',
(F,F){r\F)= [(Fr)-^{Fir)]K
Therefore - ^^^^^ _ + — (^1^') - - = + 1
inereiore ^|(j?.|^) (^/ 1^)} + ^i(F\F) (F' \F')} "
Similarly from the equation
\(F-\F) + \'{F'-\F') = 0,
we deduce
= + 1.
^{(F \F) (F' , F')\ ^{{F I ^) (^' I r)}
But it has been proved in § 231, using its notation, that
^ ^ ^ = cos - cos — , - rojp-iri\7hn-ff7vi = ^^^ ~ ^^^ ~ •
^{(F\F){r\F')] --y--y> ^{{F\F){r\r)} "'"7 7
Hence for right-parallelism, assuming that — and — are both acute
angles but not necessarily both positive (with the usual conventions as to
signs of lengths), cos Si cos Sj + sin Sj sin S^ = 1 ; therefore Si = Sa-
For left-parallelism, cos Si cos Sg — sin Sj sin Sg = 1 ; therefore Si = — S,.
But Si and Sj taken positively are the greatest and least perpendicular
distances from one line to the other. Hence the lines are parallel according
to § 229.
(4) Thus through any point 6, a right-parallel line and a left-parallel
line to any line F may be obtained by the following construction.
Draw ba perpendicular to F, and let the least of the two distances of 6
from -P be S, which is ba. Find the polar line of ab, which must intersect F
at right-angles in some point o. On this line take d and df on opposite sides
of c at distance S from it. Then bd and 6d' are respectively the right and
left-parallel to F through 6. It is to be noted that \F is both a right
and a left-parallel to F; and that a line parallel to i^ is parallel to \F.
(5) Since two parallel lines are generators of the same quadric [cf. sub-
section (2)], they are not coplanar.
234, 235] VECTOR SYSTEMS. 407
236. Vector Systems* (1) Any system (8) of the type F±\F ia
called a vector system. Such a system has an infinite number of pairs of
axes, consisting of all lines parallel to F taken in pairs. For let F' be
any right or left-parallel to F. Then a relation exists of the form,
F±\F^\(r± \r). Accordingly flf = X (^' ± \Fy
Let a system of the form i^H-|-Pbe called a right- vector system. If 12
be a right- vector system, jB = |jB, and (R R) = (BR): either of these equa-
tions is a sufficient test, if the system is known to be real [cf. § 233 (4)].
Let a system of the form i^— |i^ be called a left- vector system. If Z be a
left- vector system, i = — (i, and (i|i) = — (iZ): either of these equations
is a sufficient test, if the system is known to be real. Vector systems are
the self-supplementary systems of § 174 (2).
(2) The sum of two right-vector systems is a right-vector system, and
the sum of two left-vector systems is a left-vector system ; but the sum
of a right- vector system and of a left- vector system is not a vector system f.
For let R = aiO^ + laiOj, and R' = 6i6a + Iftitj, be two right- vector systems.
Then R-{-R = (aiO, 4- 6i6a) + | (aiO^ + bA).
Now let diO^ + 6162 = CiCa + € I CiCg.
Then 22 + iiT = (1+ e) (CiC+ \cc^).
Accordingly 12 + 22' is a right- vector system.
A similar proof shows that the sum of two left-vector systems is a left-
vector system. It is also obvious that another statement of the same
proposition is that the dual group defined by two vector systems of the
same name (right or left) contains only vector systems of that name.
(3) But if 12 is a right-vector system and i is a left-vector system,
then 12 -h i is not a vector system.
For if it were, 12 + Z = |(12 + i) = 12 - i.
Hence R = L. But a system cannot be both a right and a left vector
system ; since for such a system, |/8=flf= — /S, which is impossible.
Any system J 8 can be written in the form 12 -h Z. For
fif = i(S + |fif)-hi(S-|fif);
and J (/Sf + 1 S) is a right- vector system, and i (/S- |S) is a left- vector system.
This reduction is unique. For if flf == 12 + i = 22' + L\ then R-R'^L-L. 1
Hence a right-vector system would be equal to a left-vector system, which is )
impossible.
* This use of the word * vector ' Beems to me to be very nnfortiinate. Bat an analogous ose ib
too weU established in connection with the Idnematics of Elliptie space to be altered now. The
theoiy of systems of forces is veiy analogons, as will, be proved later, to the theory of motors and
vectors investigated by Clifford ; cf. toe. cit.^ p. 870.
t Cf. Sir R. 8. Ball, "On the Theory of Content," Boyal Irish Academy, Trantactiofu, 1889.
X Cf. Clifford, loc. cit, p. 370.
{
408 EXTENSIVE MANIFOLDS AND ELLIPTIC GEOMETRY. [CHAP. IIL
(4?) Any right- vector system and any left- vector system are reciprocal*.
For let R = OiO, + 1 OiCt,, and L = 6163 — | bfi^, then (RL) = 0.
236. Vector Systems and Parallel Lines. (1) Let e^ and e^ be
two unit quadrantal points : then the vector systems CiC^ ± [eiC^ are called
unit vector systems. If R= e^e^ + l^iej, then
{RR) = 2 {e^e^ \e,e^) = 2 {e, k) {e^ |e,) = 2 = (iJ IE).
Also if i = ei^j — lei^a, then (Zi) = — 2 = — (i |i).
(2) A vector system -f- possesses an infinite number of axes, it being
possible to draw an axis of the system through any point; and this set
of axes forms a set of parallels, right or left according as the vector
system is right or left.
For let X be any other unit point and let p be another unit point on the
right parallel to e^e^ and quadrantal to x.
Then by § 234 (2), jB = ^leaH- |eiej = X(icpH-|icp). Hence xp and \a^p are
also axes of R.
Also (jB|i2) = 2 = 2Xl Hence X=+l. Thus a unit vector system,
when expressed in terms of one pair of axes, is a unit vector system when
expressed in terms of any other pair of axes.
(3) A simple expression J for a line drawn through a given point right or
left-parallel to a given line can be found. For let a^a^ be the given line and x
the given point. Consider the right- vector system R = Oia, + loiO,. Let xp
be the required right-parallel to aia^ drawn through x. Then
iJ=\(a?pH- \xp).
Hence xR = \x \xp = X (a? |p) |a? — \ (a; \x) \p ;
therefore x \xR = — X (a? | a?) a?p = a?p.
Hence x\xR is a force on the right-parallel to OiOa drawn through x,
where R = OiOa + {oiO^,
Similarly if Z^aiOs — lajOa, then x\xL is the left-parallel to aiCt, drawn
through X,
(4) It follows that, if any two lines are each parallels of the same name
(right or left) to a third line, they are parallels of that name to each other.
Let all the lines parallel (of the same name) to a given line be called a
parallel set of lines.
(5) Any pair of conjugates of a vector system is a pair of parallel lines
of an opposite denomination (right or left) to that of the systemj.
* Cf. Sir R. S. Ball, Transactions R,LA,
+ Cf. Clifford, loc, eit., p. 370.
t Not previously published, as far as I am aware.
236, 237] VECTOK systems and parallel lines. 409
For let R = \F+fiF\
Then R=^\F+ fiF' = \R = \\F-{^fi\F'.
Hence X (i^- 1 J^ + m (F - \F') = 0.
Thus by § 234 (2) F and F' are left-parallels, while i? is a right-vector
system.
It is to be noted that by § 234 (4) any pair of axes of a vector system,
since they are reciprocally polar lines, are both right and left-parallels.
(6) Any* right-parallel set of lines and any left-parallel set of
lines have one and only one pair of reciprocally polar lines in common
[cf. § 234 (4)]. For let R and L be the associated vector systems of the two
sets of parallels. Then they are necessarily reciprocal ; also they have
one and only one pair of common conjugates. These common conjugates
are the lines F and F\ where
F=R ^{LL) + i v{- {RR)] ; F'^R ^{LL) - i V{- {RR)Y
Hence F' = \F.
Also R = {F^-\F\ i = (F-|-F).
Thus F and F belong to the right-parallel set of R and to the left-
parallel set of L,
(7) The common conjugates of two vector systems of the same denomi-
nation are a pair of imaginary generating lines of the absolute. This follows
from § 235 (2).
237. Further Properties op Parallel LiNEsf. (1) If any straight
line meet two parallel straight lines, it makes each exterior angle equal to
the interior and opposite angle, or in other words the two interior angles
equal to two right angles.
For let xp and yq be two parallel lines (say right-parallel) ; and let ay
be any line meeting them. Then pxy, and qyx are the two interior angles.
Fio. 7.
Let X, y, p, q be all unit points, and let (x |p) = 0 = (y \q).
Then, by § 236 (2), it may be assumed that
a?p+|icp = y? + ly? (1).
* Not previously pablished, as far as I am aware. t Gf. Clifford, Uk, cit. , p. 370.
410
EXTENSIVE MANIFOLDS AND ELLIPTIC GEOMETRY. [CHAP. III.
Also
Similarly
cos Qp = (^y^) = (^y 1^/^).
cos Qj = ^y^^M) = - Jp^y\m)
^^ ^J[{^ \3oy) (yq \yq)} ^/[xy \xy] '
But from equation (1), multiplying by ayy, we find {xy xp) = (xy yq^-
Hence cos yxp + cos xyq = 0, and yxp + xyq = tt.
A similar proof applies to left-parallel lines.
(2) Conversely, if two straight lines be such that every line intersecting
both makes the two interior angles equal to two right angles, then the liaes
are parallel.
For let any line a^cut the lines xp^yq; so that yxp = xyq\.
Fig. 8.
Draw xl and yvi perpendicular to yq and xp respectively.
Then from § 216, considering the triangles ooyl and xyrriy
. xl , xy . ym
sm — sm -^ "'•* ^
_^ = _j:=sin?^
7
sm
sm xyq sm ^ ' sin yocp
Hence xl = ym.
Therefore the lines ocp and yq are parallel.
(3) Parallelograms can be proved to exist in Elliptic Space: but they
are not plane figures [cf. § 234 (5)].
For let oh and ac be any two lines intersecting at a. Then the right-
Fio. 9.
237] FURTHER PROPERTIES OF PARALLEL LINES. 411
parallel through b to etc is, by § 236 (3), F=bb(ac+] ac). Similarly the
lefb-parallel through c to ab is F' = c\c (ab — \ah).
To prove that these lines intersect, we have to prove that (FF') = 0.
But it is easy to prove by multiplication and reduction that,
{FF') = b {6(ac + |ac)}.cl{c(a6-|a6)} = 0.
Therefore the two parallels through b and c intersect in some point d.
Therefore the opposite sides of the figure abdc are parallel, one pair being
right-parallels and the other pair being left-parallels.
Also if the angle cab be d, then abd = w^0, bdc = 0, dca = tt — ft
Further it is easy to prove that ab=cd, and bd = ac. Thus the opposite
sides are equal. Hence if ac and bd be any two parallels and ac = bd, then
oi and cd are parallels of opposite name (right or left) to ac and bd\ and
also aJ)=^cd.
(4) Let ab, ab' be one pair of parallels, and let (W, a'c be another pair
of the same name as the first pair: also let at = q!V, ac = a'c', then be and Vc
are parallels of the same name, and be = b'c'. For join oaj!, bV, cc\
Then by (3) aa' and 66' are equal and parallels, of the opposite name to
a6 and a'6' ; also aa' and cc' are equal and parallels of the opposite name to
ac and old. Hence [cf. § 236 (4)] 66' and cc' are equal and parallels of the
opposite name to a6 and a'6'. Hence 6c and 6'c^ are equal and parallels of the
same name to a6 and a'6'.
Fig. 10.
It is further obvious that the angle ca6 is equal to the angle cV6'.
Hence if fix)m any point a' two parallels of the same name are drawn
to any two lines a6, ac, the two pairs of intersecting lines contain the same
angle.
412
EXTENSIVE MANIFOLDS AND ELLIPTIC GEOMETRY. [CHAP. IIL
238. Planes and Parallel Lines*. (1) One line, and only one line,
belonging to a given parallel set of lines, lies in a given plane.
For let P be the given plane and exS^ a line of the given parallel set.
Now if F be one of the set lying in P,\F is also one of the set
and passes through the point \P ; and conversely. But one and only one of
the set can be drawn through |P, hence one and only one of the set lies
in P.
li p stand for |P, then by § 236 (3) the right-parallel |P, which passes
through \P/\Bp \pRy where jB stands for eye^ + '^e,. Hence
F=p.\.\p\R^P\PR.
Thus the single right-parallel in the plane P is the line P|Pi2. Similarly
for left-parallels.
(2) To any point jp in a given plane P there corresponds one and only
one point q in any other given plane Q, such that if any line through p be
drawn in the plane P, the right-parallel line through q lies in the plane Q
(or in other words the right-parallel in the plane Q passes through q).
For draw any two lines pp\ pp" in the plane P. Let their right parallels
in the plane Q be qq[, qq" intersecting in g. Then q is the required point.
For take pp = qj^, pp' = qjq[\
Then pp' + W=-qq' + qq\ pp^-^W' = ??" + I??"-
Any other line through p and in the plane P can be written pp' -h \pp".
But from the above equations,
pp' -I- \pf -h 1 {pp' -h \pp'') ^q^^ -Kqf + \(qq' + \q^').
Hence the line qq + X^g", which passes through q and lies in the plane
Q, is the right-parallel to the line pp' + Xpp".
Fig. 11.
* These properties have not heen stated before, as far as I am aware.
238]
PLANES AND PARALLEL LINES.
413
Similarly a unique point qi in the plane Q corresponds to the point p
in the plane P with similar properties for left-parallels.
(3) With the construction of the preceding proposition (where ^' = qq',
PP'^W\ i* follows from § 237 (4) that p'p" is the right-parallel to qq".
Hence the points p' and j' in the planes P and Q correspond. Thus, given two
corresponding points p and q^ it is easy to find the point on one plane corre-
sponding to any point on the other. For consider the point p' on P. Join
pp' and draw qq' parallel to pp' and of the same length. Then q corresponds
to p\
(4) The common perpendicular of two planes P and Q, namely |PQ, cuts
the planes in two points p and q which are corresponding points both for
Fig. 12.
right and left-parallels. For in the plane P dmw any line pm cutting the
line PQ in m. Take two points I and r on PQ such that lm = mr—pq.
Join qr and ql Then [cf § 234 (4)] one of them (say qr) is a right-parallel
to pm and the other g! is a left-parallel.
Accordingly, knowing that p and q are corresponding points, it is possible
by (3) to construct the points ji and q^ on Q corresponding to any point p' on
P for right and left-parallels respectively.
CHAPTER IV.
Hyperbolic Geometry.
239. Space and Anti-space. (1) In hyperbolic geometry [cf. § 208]
the absolute
is a real closed quadnc.
If 6 be any point within such a quadric, then [cf. § 82 (6)] its polar plane
does not cut the quadric in real points and the polar plane lies entirely
without the quadric. Hence if ^, ^i, ^g, ... e^i form a normal system, and if
e lie within the quadric, then the remaining points Ci, e^, ... a^-i lie without
it. Similarly [cf. § 82 (7)] if E, Ei, E^, ... ^^-i fonn a normal system of
planes, and if E does not cut the quadric in real points, then Ei, E^, ... E^i
must all cut the quadric in real points and include points within the quadric.
(2) Let that part of the complete spatial manifold of i^ — 1 dimensions
which is enclosed within the absolute be called Space [cf. § 202]. Let the
part without the absolute be called Anti-space, or Ideal Space. Let a point
within space be called spatial, a point in anti-space anti-spatial.
(3) A subregion may lie completely in anti-space, as far as its real
elements are concerned, but cannot lie completely in space. Let a subregion
which comprises spatial elements be called spatial, and a subregion which
does not comprise spatial elements anti-spatial.
(4) Then a normal system of real elements e, ei, ... e^-i consists of one
spatial element e, and of i^ — 1 anti-spatial elements. Let e be called the
origin of this system.
A normal system of planes E, Ei, ... E^i consists of one anti-spatial
plane E, and of i^ — 1 spatial planes.
If a plane P be spatial, its absolute pole \P is anti-spatial; if the plane
be anti-spatial, its absolute pole is spatial.
If an element p be spatial, its absolute polar \p is anti-spatial ; if p be
anti-spatial, \p is spatial.
If any subregion Pp, of p — 1 dimensions, be spatial, the subregion [P^ is
anti-spatial; if Pp be anti -spatial, |Pp is spatial.
239, 240] INTENSITIES OF POINTS AND PLANES. 415
240. Intensities of Points and Planes. (1) Let the absolute be
referred to the v normal elements e^Bu ... «ir-i, of which e is spatial.
Let a, itti, ia ioLp-i be the normal intensities [cf. § 110] of these
elements; and let i*^*A stand for i*^^aaiaj... a^-i. Then A is also real,
where a, Ci, ... a„_i are real.
Also let v^^ A {eei . . . 6^-1) = 1.
t»^i A — t*^' A
A nen | b ^ r B\B^ ... 6y— .1 , ( ^ ^ o BB2 • • . By — 1 ,
a — ofi
|e2 = T- ^1^5 . . . By-i ; and so on.
Hence if a? == f e + f i^i + . . . + f r-i^r-i >
then (^|^)=(^«Il;«^;....«&z.A
Thus [cf. § 82 (9)] if a? be spatial and its co-ordinates real, (a? |a?) is positive.
This supposition will be adhered to.
(2) Any real plane is given by
X/ ^^ \B\B2 • • • Bff-^\ """ A/iBB% . • • Bp—m\ "T" ht^fi^^o% ... By—\ ~r etc. ^
where the ratios X : Xi : X2 : etc. are real.
But if X (= X^b) be a real point with its co-ordinates real, we may suppose
X to be the pole of i, and write i = |a?
= t*""* A f -J eiCj ... Br-i - -^ eea ... ^^-i + etc.) .
Hence \ = i»^i_?, x^==i'-i— i^ etc.
Therefore if j; be even, X, X^, etc., are pure imaginaries, so that their
ratios are real.
A plane will be considered to be in its standard form, when expressed in
the form
Ju ^^ % KSiB^ ... By—\ ~- % A^^^s ... By 1 "T" etc.,
where X, Xi, etc. are real.
Then we can write L = |a?, where the coefficients of x are real. Thus a
real plane is, if v be even, intensively imaginary [cf. § 88 (3)] ; while a real
point, spatial or anti-spatial, is always intensively real.
(3) If a; be spatial, its intensity is unity when (a; |ar) = 1, and is real
when (a; I a?) is positive.
If a? be anti-spatial, the intensity of x will be defined to be real when
{x I a?) is negative, and to be unity when (a? |a?) = — 1.
Thus in both cases the intensity is real when the coefficients are real.
416 HYPERBOLIC GEOMETRY. [CHAP. IV.
The intensity [cf. § 211 (5)] of a plane L will be defined to be unity,
when it is in the standard form \x, where a; is at unit intensity.
Thus for anti-spatial planes at unit intensity, \x is spatial, and
(i|i) = (|a?.||a?)=(a?|a?)=l.
For spatial planes at unit intensity
(Z|i)=-1.
(4) Thus for a spatial point 2fe at unit intensity
For a spatial plane Xi^^ \E at unit intensity,
(5) But if the reference elements e, Ci, ... e„^i be at unit intensities,
spatial and anti-spatial, then a = Ai = ... = o^-i = 1.
Hence [cf. subsection (1)] the point-equation of the absolute is
P ~ f 1 • . • f y—l ~ ">
the plane-equation of the absolute is
Ani "T" A.2 I ••• "i" A» y.^1 -~ A» =^ U.
The intensity of a spatial point 2fc is (f— fi' — fa*— ... — ^k-i)*; the
intensity of a spatial plane 2i"^* \E is (Xi* + X,* -h . . . -h X Vi — X*)*.
Also i'~' (eei ... e^i) = 1 ;
and 1 6 ^^ l ^i^a • • • ^r — 1 > I ^1 ^" ^ €€2 . • • €^ — 1 f 1 ftj ^^ '^ 1 BB\€^ • . • 0|f — 1 ,
I Cj = t'"""eeie4 . . . e^i, and so on.
This supposition will be adhered to, unless it is otherwise stated.
241. Distances of Points. (1) It will be seen that in the case, in
which the line joining two points in anti-space does not cut the absolute in
real points, the usual hyperbolic formula does not give a real distance between
them. In this case it is convenient to use the Elliptic measure of distance.
Thus any two points in anti-space (as well as any two points in space) are
separated by a real distance. Elliptic or Hyperbolic. But a point in space
cannot have a real distance of either type from a point in anti-space.
(2) Firstly, let two points x and y both be spatial, and of standard sign
[cf. § 208 (3)] ; then xy is given by
7 Vl(^k)(y|y)}'
and ^, thus determined, is real. Also, since there can be no distinction
between D (xy) and ^, the latter symbol will always be used for the distance.
241, 242] DISTANCES OF POINTS. 417
(3) Secondly, let w and y both be anti-spatial, but let the line xy be
spatial. Then if Oi and a^ be the points where the line meets the quadric,
X and y lie together on the same intercept between Oi and a^. Also (^|^)
and (y \y) are of the same negative sign. Hence the hyperbolic functions
give a real distance. Thus xy is determined as a real quantity by the
formula
cosh ^ = — ^^M__ •
7 V{(«?l«)(y|y)}'
where the ambiguity of sign must be so determined that the right-hand side
is positive.
(4) Thirdly, let x be spatial and y be anti-spatial. Then both the
formulae
cosh^_ (^ly)
cos 7 ~V{(«k)(yly)}
must make ^ imaginary, since {x \x) and (y \y) are of opposite signs.
(5) Fourthly, let x and y both be anti-spatial, and let the line xy be
anti-spatial. Then the two elements Oi and a^ in which ay meets the absolute
are imaginary. Hence the elliptic law applies. Let the distance between
X and y, determined by this law, be called the angular distance between the
points, and denoted by Z xy.
Then cos Z aw = ... ^^}y — rr ;
where the conventions of § 206 apply : so that, if a?' stand for — x,
.^ W\y)
cos Z ary = , .. ,. \f, . v^ = — cos Z xy.
Hence Z x'y •\- /.xy^ir,
(6) Thus, to conclude, if the line xy be spatial, and x and y be both
spatial or both anti-spatial, then ^ is real. If the line xy be anti-spatial,
then jl xy is real. If x be spatial and y anti-spatial, both ^ and Z xy are
imaginary.
242. Distances of Planes. (1) Consider the formulae for the separa-
tion between two planes P and Q.
Firstly, let both planes be spatial, and let the subplane PQ be spatial.
Then |P and |Q are both anti-spatial, and |P |Q is anti-spatial.
Hence Z |P |Q is real, and is given by
cosz|P|Q = ^^^^^> = ^^l^>
coszi^-iv ^{(|i>||P)(|Q||0} ^[(P\P){Q\Q)y
Hence the separation between P and Q is real, when determined by the
elliptic formula. Let it be called the angle between P and Q, and denoted
by /.PQ.
w. 27
418 HYPERBOLIC GEOMETRY. [CHAP. IV.
Then co8^PQ=-^p^^^.
It is to be noticed that there are two angles Z PQ and tt ~ Z PQ, corre-
sponding to the ambiguity of sign on the right-hand sida
(2) Secondly, let both planes be spatial, and let the subplane PQ be
anti-spatiaL Then \P and \Q are both anti-spatial, and |P|Q is spatial.
Hence |P |Q is real, and is determined by
_ (P\Q)
- V{(P |P) (Q 10} •
Hence the separation between P and Q is to be measured by the hyper-
bolic formula, and will be called the distance between the planes, and denoted
by PQ. _
Then oosh^g^ (^IQ)_.
Ihen cosn ^ -^{(P|P)((2|g)}.
where as usual the terms P and Q are so chosen that (P | Q) is positive.
(3) Thirdly, let P be spatial and Q be anti-spatiaL Then (P|P) and
(Q \Q) ^^ ^f opposite signs. Hence both Z PQ and PQ are imaginary.
(4) Fourthly, let the planes P and Q both be anti-spatial. Then \P
and \Q and \P \Q are spatial, and |P {Q is real.
Hence cosh i^i^ = — i^?M__
Hence cosh ^ " VK|P ||P) (|Q||Q)}
_ (P|Q) •
"VKi'lPXQIQ)}-
Hence PQ is real and Z PQ is imaginary.
Ai ^.PQ_ (P|Q)
7 ~VK^|-P)(Q|<2)}'
where the terms P and Q are so chosen that (P | Q) is positive.
243. Spatial and Anti-spatial Lines. (1) If the elliptic measure
for separation holds, then [cf. §§ 204 and 211]
sin Z aw = A / ^f^Jf^\. ,
and sinzPO- /_l^i™_ -
and if the hyperbolic measure holds, then [cf. §§ 208 and 211]
ainh ^ - / -(^1^)
7"VK^k)(y|y)}'
and sinh
PQ I -(PQ\PQ)
V
7 V{iP\P)(Q\Q)V
1
243, 244] SPATIAL AND ANTI-SPATIAL LINES. 419
(2) Thus if 0^ be spatial, {pcy \asy) is negative. For if x and y be either
both spatial or both anti-spatial, the proposition follows from the expression
for sinh — . But if x be spatial and y anti-spatial, then
(xy \xy) = {x \x) {y\y)-{x |y)".
Now (x\x) is positive and (y\y) negative; hence again the proposition
follows. But if xy be anti-spatial, then, from the expression for sin/lxy,
(sy\xy) is positive.
(3) Furthermore if x be anti-spatial and y be any point on the cone,
(xy \(vy) = 0, which envelopes the quadric, then xy = 0 and Z icy = 0. Hence
any two points on a tangent line to the quadric are at zero distance from
each other.
(4) Again, by similar reasoning, if the intersection of two spatial planes
P and Q be spatial, {PQ \PQ) is positive. If the intersection of two spatial
planes be anti-spatial, (PQ \PQ) is negative. If P be spatial and Q anti-
spatial, (PQ \PQ) is negative. Hence if PQ be spatial, (PQ \PQ) is positive ;
if PQ be anti-spatial, (PQ \ PQ) is negative.
244. Distances of Subregions. (1) If two subregions Pf, and Qp,
each of /D — 1 dimensions, are contained in the same subregion (L) of p
dimensions, then [cf. § 211 (6)] a single measure of the separation of Pp
and Qp can be assigned.
(2) Let the section of the absolute by L be real; and firstly let the
intersection of Pp and Qp be spatial. Then Z PpQp is real, and
cos z P 0 = ^'^ IQ**^ ,
COSZi^pWp V{(pjp^)(QJQ^)}-
Secondly, let the intersection of Pp and Q^ be anti-spatial, but P^ and Qp
be both spatial. Then P^Q^ is real, and
'"^ 7 "VKi^pli^pXQplCp)}'
Thirdly, let P,, be spatial and Qp be anti-spatial. Then P^Q^ and Z P^Q^
are both imaginary.
Fourthly, let P^ and Qp be both anti-spatial. Then P^Q^ is real, and
7 ~'/{{p,\p,){Q,mv
(3) Let the section of the absolute by i be imaginary. Then P^ and Q,
are anti-spatial, and we have a fifth case when Z. P,Q, is real, and given by
the formula of the first case.
27—2
420 HYPERBOLIC GEOMETRY. [CHAP. IV.
246. Geometrical Signification. Geometrical meanings can be
assigned to the co-ordinates of any spatial point x, at unit intensity
and of standard sign, referred to a normal system «, e^, ... e^-i at unit in-
tensities, of which e is the spatial origin.
Let a7=rfe + fiei+ ... + ?,^A-.i, where (a?|a?) = l = f — fi- ... — fr-i*
Then
cosh
Let the
angles,
a?ec,
= Xl:
, xee2
r
Then
COS
jX,=
{esr^ex)
*s/[{ex ex)(eei eei)}
ss
. , ex
Rinh —
Hence f , = sinh — cos \,, f, — sinh — cos X^, etc.
Also the angles X,, X^, ... X,^i are connected by,
2cos"X = l.
»
Similar geometrical interpretations hold for Elliptic Geometry.
246. Poles and Polars. (1) It will be noticed that the only case,
when there is no real measure of separation between two points x and y,
is when x is spatial and y is anti-spatial. In this case the point of intersec-
tion of xy and the polar of y is spatial. For this point is
xy\y = (x y)y-Q/\y)x=:y',sa.y.
Then by simple multiplication we find
(i/ \y)='(y \yy (« 1^)- («? lyYiy^y)-
But (x x) is positive and (yiy) is negative. Hence (y' \j/) is positive,
and therefore y' is spatial.
Also the term y' is of standard sign. For [c£ 208 (8)] a? is by h)rpothesis
of standard sign, and
(x \y') ^(x\yy^(x\x) {y \y) ^-{xy \xy),
' But xy is spatial; hence, by § 243 (2), {xy\xy) is negative, and {x jf)
is positive.
Similarly the point of intersection of xy and the pol^ar of a: is a?' = yx \x ;
and x' is anti-spatial, since \x is anti-spatial.
(2) Now (y' \i/) = (y \y) {{x \x) {y \y) - {x |y)»} = (y \y) {xy\xy\
and (y ' k) = (a? | y )» - (a? ; «;) (y | y) = - (ajy i xy).
Hence cosh ^ = ../y... . = . A ^i"f , •
245, 246] POLES and polaks. 421
Also since (y|y), as well as (oi!y\ay\ is negative, j/x is real as given by
this formula.
Similarly sinh ^ = a / ] cosh* ^ - 1 [
^{-(oo\x)(y\y)}'
Also since x' and y are both anti-spatial, and afy ia spatial, then a/y is
real. And by a similar proof
cosh^=.A-(^^, = cosh2^.
Hence a/y = y'a?.
(3) Let X and y be both anti-spatial ; and let ocy be spatial. Then ^ is
real.
Also x'=^yx\ Xy and y' = xy\y are both spatial points. For
(y' |y') = (y |y)" («? 1^) -{^\yyQ/\y)={y |y) (^ 1^)-
Now (y |y) and (a7y|«y) are, by hypothesis, both negative.
Hence (y' |y') is positive, and y' is a spatial point. Similarly a?' is a spatial
point.
Also (x' ly') = (x \yy - (x \x){y \y) {x\y)=^-{x \y) {xy \xy).
Hence if the terms x and y be so chosen as to sign that (a? |y) is positive,
{x ly') is also positive.
Now, since af and j/ are both spatial, afy^ is real ; and
nn«>, ^V. iP^'W) -(a?|y)(ay|d?y)
'""'^ 7 "VKI^'Xy'lyOl' + VK^I^XylyX^l^y)"}*
Now, since (xy \xy) is negative,
+ y {(^ 1^) (y |y) (^ ky)"} = - (^^ l«^) V{(a? 1^?) (y |y)}.
= cosh — .
7
(4) Exactly in the same way let the plane P be spatial and Q anti-
spatial, then \P is anti-spatial and \Q is spatial. Let the plane through PQ
and I P be called F and that through PQ and \Q be called Q'. Then Q! is
obviously spatial, and P' can be proved to be anti-spatial.
Then P and Q are two spatial planes with an anti-spatial intersection.
Hence cosh — = ± ,JJpYp)^^\qy^ •
But Q' = PQ|Q = (P|Q)«-(QiQ)i'-
Hence
7 V{(^ ^) (y y)}
Therefore
a: V = a^-
422 HYPERBOLIC GEOMETBT. [CHAP. IT.
Therefore (P|Q') = (P|Q)»-((2|Q)(P|P) = -(PQ|P(2),
and {Q'm^{Q\QHPQ\PQy
Hence cosh^'- / JlQ}m . ^ ,osh^
uence cosn ^ ~ V (i'li'XQIQ)"" 7 '
Q- 1 1 -.K^ (P|Q) . , FQ
Similarly sinh — = ^|, (j> |J./(q|Q)j = srnh — .
Hence PQ ^FQ,
247. Points on the Absolute. (1) A point u on the absolute is at an
infinite distance from any other point. For (u\u)^ 0, and hence
cosh— = . ; / . = « .
(2) To find a point u in which any spatial line xy cuts the absolute,
put w = a? + Xy.
Hence X»(y \y) + 2\{x \y) + (a? |ic) = 0.
Now let p stand for the distance xy^ and let x and y be spatial points at
unit intensity and of standard sign.
o t JL
Hence X" + 2Xco8h^ + l = (X + ey)(X + 6 >) = 0.
Accordingly the two points, in which the line xy cuts the absolute, are
x — e^yy and x — ^y.
(3) In the same way if the line xy be spatial, but x and y be both anti-
spatial at unit anti-spatial intensity, and {x \y) be positive, then the points,
in which osy cuts the absolute, are x + e ^ y,x-\-& y,
(4) Similarly let P, Q be two spatial planes, and PQ be anti-spatial :
also let P and Q be at unit spatial intensity and let p be the distance
between them (hyperbolic measure). Then the planes through PQ touching
-t t
the absolute are P-\-e y Q, and P + e^r Q.
Also if P and Q be both anti-spatial at unit anti-spatial intensity, the
tangent planes are P — e"y Q, and P — ^ Q.
248. Triangles. (1) Consider a triangle o&c, in which the measures
for the separation of the angular points are all real. Then the cases which
arise are (1) a, 6, c all spatial ; (2) a, h, c all anti-spatial, and be, ca, afr all
spatial ; (3) a, b, c all anti-spatial, and be, ca, ab all anti-spatial ; (4) a, b, c
all anti-spatial, and be, ca, ab two spatial and one anti-spatial ; (5) a, 6, c all
anti-spatial, and be, ca, ab being one spatial and two anti-spatial.
247, 248] TRIANGLES. 423
(2) Case /. a, 6, c all spatial. Let the triangle abc in this case be
called a spatial triangle.
Let the angle between ah and ac be a, that between ha and bche jS, and
that between ca and cb be y.
To discriminate between a and tt — a, let a be that angle which vanishes
when b coincides with c ; and similarly for J8 and y.
Thus, b and c being of standard sign according to the usual convention,
_ {ab \ac)
^ ^/{{ab \ab) {ac \ ac)} '
And (ab |ac) = (a \a) {b\c) -(a\b) (a \c) ;
I . , oft / — (afclfltft) . ,ac / — (ac|ac)
also sinh — = a / / i x/lil\ » sinh — = a / / i A/ i\ •
7 V(a|a)(t|&) 7 VJa|a)(c|c)
, 6c , oft , ac
cosh cosh — cosh —
Hence coea = — "^ _ 1^ _ 7 .
. , a& . . ac
sinh — sinh —
_ _ jy !_ _
•n,. II ,6c 1 a6 , ac . , a6 . , ac
r mall y, cosh ■— = cosh — cosh smh — sinh — cos a.
7 7 7 7 7
(3) Also [cf. § 216 (1)]
Din a = "^^^^ 1^^ (oc |ac) - (a6 lacy}
\/{(a6 |a6)(ac|ac)}
_V{(a|a)(a6c|o6c)}
"V{(a6|a6)(ac|ac)}'
And ^^^^ = Jm^v
Therefore -2i2iL = /(a|a)(6|6)(c:c)(a6c |a6c)
. V 6c V - (6c i6c) (ca lea) (a6 |a6)
sinh —
7
_ sin jS siny
sinh^ sin?5^'
7 7
(4) It is easily proved, exactly as in Elliptic Gteometry [cf § 216 (6)],
that the perimeter of a spatial circle, with a spatial centre and of radius p, is
iiry sinh - . And that the length of an arc subtending an angle a at the
centre is a7 sinh - .
7
(5) Case II, The angular points a, 6, c are anti-spatial, and the sides
5c, cay ab are spatial. Let the triangle a6c in this case be called a semi-
spatial triangle.
424
HYPERBOLIC GEOMETRY.
[chap. it.
The distances between the sides be, ca, ab must now be measured by the
hyperbolic measure. Thus let a, yff, y be assumed to be lengths and not
angles. Also adopt the conventions of Case I. Then by a similar proof
,6c ,a6 ^ ac . <. ah . , ac i_flt
cosh — = cosh — cosh smh — smh — cosh — .
7 7 7 1 1 y
sinh - sinh ^
Also
. , 6c . . ca . , a6
smh — smh — sinh —
7 7 7
_ I {a I a) (b 1 6)^c ( c) (a6c ! ahc)
" V ~(fio\ 6c)^ca |ca) (ab\ab)
(6) Case HI. The angular points a, 6, c all anti-spatial, and 6c, ca, oh
also anti-spatial.
This case gives simply the ordinary formulae of Elliptic Geometry [of. § 215].
(7) Case IV, The angular points a, 6, c are anti-spatial, the two sides
ah, ac are spatial, and the third side 6c is anti-spatial.
The sides ab and ac have a real measure of separation a, reckoned
according to the hyperbolic formula, but the sides ba and 6c, and the sides
ca and c6 have no real measure of separation.
TT . i i_ct6 iCtc .,a6.,ac ,a
Hence cos ^oc^ cosh — cosh smh — smh — cosh - .
7 7 7 'y . *>"
(8) Ca^se V. The angular points a, 6, c are anti-spatial, and ab, ac sie
anti-spatial and 6c is spatial. Then a is real, and JS and y are imaginary ;
and a is measured by the elliptic formula. And
6c
cosh — = cos Z a6 cos Z oc + sin Z a6 sin Z oc cos a.
7
There are no corresponding formulae to be obtained by cyclic interchange;
since ^ and y are imaginary.
(9) The theory, given in § 217, of points inside a triangle holds without
change for Hyperbolic Geometry.
249. Properties of Angles op a Spatial Triangle. (I) Two
angles of a spatial triangle cannot be obtuse. For if a and JS be both
obtuse, cos a and cos JS are both negative. Hence from § 248 (2)
ac
^ ab ^ ac t be , ,6c ,ct6 iw/
cosh — cosh — < cosh — , and cosh — cosh — < cosh — .
_ 7 7 7 7 7 7
But cosh — is necessarily greater than unity; hence these two inequalities
are inconsistent. ^
\
ft:-'
249, 250] PKOPERTIES OF ANGLES OF A SPATIAL TRIANGLE. 425
(2) It follows from § 247 (2) that, when b and c are spatial points of
standard sign, all points of the form \b + /ac, where \//a is positive, lie on the
intercept between b and c ; since the two points, in which xy cuts the absolute,
both lie on that intercept for which \/fi is negative. Hence it may be proved,
exactly in the same way as in § 219 (2) dealing with Elliptic Geometry, that
if in any triangle abc JS and y be both acute, the foot of the perpendicular
from a on to 6c falls within the intercept be.
(3) The sum of the angles of any spatial triangle is less than two right-
angles.
Firstly, let the angle y be a right-angle. Then as in Elliptic Geometry,
[cf. § 219 (4)].
1 6c ~ ca
cosh 5
tanH« + A) = , ^ 'y — cot j^Y
or^h 1 fee + CO
2 7
tlbc^ca
cosh H
Ljl^.
tlbc+ca
cosh 2;
2 7
XT , 1 6c - ca ,lbc + ca
Now cosh 5 < cosh 5 .
27 27
Hence a + yff < |^ .
Hence a + yff + y < tt.
Secondly, the theorem can. be extended to any triangle by the reasoning
of § 219 (4).
260. Stereometrical Triangles. (1) It is obvious by the theory of
duality that a complete set of formulae for stereometrical triangles [cf § 222 (1)]
can be set down, and that these can be ranged under eight cases just as in
the case of ordinary triangles. It will be sufficient to obtain the results for
the two most important cases.
(2) Firstly, let the planes A, B,Ohe spatial, and let the subplanes BC,
CA, AB be also spatial.
For shortness put BC = A^, CA = A, ^C'= G^.
Let ^BG = a, ZGA=JS, Z^5 = y. Also Z^id, /lC,Ai, ZulAare
real.
Then if \A = a, \B = b, \G =^ c, the triangle abc is anti-spatial, and be, ca^
ah are anti-spatiaL Hence from § 248 (6)
cos Z 6c = cos Z oft cos Z ac + sin Z oft sin Z oc cos Z (ah) (a^)).
426 HYPERBOLIC GEOMETRY. [CHAP. IV.
But z6c = i^50=a, Z.ca^ ^OA^fi, ^cib- ^AB^y. Also
Hence, cos a = cos J3 cos y + sin /? sin y cos Z Bfi^ .
(3) When the complete region is of two dimensions, this does not agree
with the ordinary formula in Euclidean space for spherical trigonometry; and,
as in the analogous case of Elliptic Geometry, the discrepancy is removed
by replacing the angles by their supplements.
When the complete region is of three or more dimensions, we deduce,
as in the case of Elliptic Geometry, that the 'Spherical Trigonometry' of
Hyperbolic Space is the same as that of ordinary Euclidean Space. This
theorem is due to J. Bolyai, &s far as space of three dimensions is concerned :
it is here extended to planes of any number of dimensions.
(4) Secondly, let the planes A^B, (7 be spatial, but let the subplanes
-^i> ^11 ^1 ^ anti-spatial. Then the triangle ahc has its three angular
points anti-spatial, but its three sides frc, ca, ah spatial. Hence from
§248(5) _____
-6c ,a6 .ac . ,ab . ac , (ab) (ac)
cosh — = cosh — cosh sinh — sm — cosh ^^ — -
7 7 7 7 7 7
Hence cosh - = cosh — cosh — — sinh — sinh — cosh —^—^ ;
7 7 7 7 7 7
with two similar formulae.
261. Perpendiculars. (1) The theory of normal points and of per-
pendiculars in Hyperbolic Geometry is much the same as in Elliptic
Geometry (cC § 223). The proofs of corresponding propositions will be
omitted.
Any two mutually normal points satisfy, (a? |y) = 0. If xy be spatial, then
one point must be spatial and the other anti-spatial [cf. § 239 (4)]. In this
case no real measure of distance exists between x and y. If xy be anti-
spatial, then the elliptic measure holds, and ^xy= \7r.
The condition that two lines a&, ao should be at right-angles (or perpen-
dicular) is {ah joo) = 0. If a be spatial, the measure of distance between the
lines is elliptic, and the angle between them is a right-angle. If a be
anti-spatial, and both lines be anti-spatial, the measure of distance is elliptic
and the angle .is a right-angle. But if a be anti-spatial, ab be spatial, and ac
be anti-spatial, there is no real measure of distance between the linea It is
impossible for two lines to be at right-angles when a is anti-spatial, and a6,
oo both spatial.
(2) If a line ab cut any region Zp, of p — l dimensions in the point a,
and if /9 — 1 independent lines drawn from a in Lp are perpendicular to ai,
then all lines dra¥na from a in Xp are perpendicular to ab.
The line ab is then said to be perpendicular to the region Xp.
251, 252] PERPENDICULARS. 427
(3) Any line perpendicular to the region Xp intersects the supplementary
(or complete normal) region jX^; and conversely, any line intersecting both
Xp and |Xp is perpendicular to both.
(4) If Pp and P„ be two regions normal to each other, and if a be any
point in P^, then any line drawn through a in the region Pf^ is perpendicular
to the region aP^*
(5) Let two planes L and M intersect in the subplane LM, and let ai be
any point in LM, From Oi draw Oql in the plane L perpendicular to the
subplane LM, and draw aim in the plane M perpendicular to LM, then the
separation between L and M is equal to that between Oil and dim.
For as in the Elliptic Geometry,
(L\M) __ (aril\aim)
^{{L\L)(M\M)} " VK«i^|OiO(aiWi|aim)} '
Hence if ZLM be real, then Ziif=Z(aif)(a,m); and if LM be real,
then LM == (ail) (aim),
(6) Any line, perpendicular to any plane L, also passes through its
absolute pole.
If any plane M include one perpendicular to L, then from any point of
the subplane LM a perpendicular to L can be dra¥na lying in M,
Also if \L lies in M, then \M lies in L ; hence this property is reciprocal.
If two planes are at right-angles, their poles are mutually normal.
(7) Also if two planes L and L' be each cut perpendicularly by a third
plane M, then the measure of separation between L and L' is the same as
that between LM and L'M,
262. The Feet of Perpendiculars. (1) Let p be the foot of the
perpendicular xp, drawn from any spatial point a; to a spatial plane L.
Then p is spatial.
For p = x\L.L] also put l=\L. Then it can easily be proved that
(p\p) = iL\L)(a>l\xl).
Now since xl is spatial, {xl\xl) is negative [cf § 243 (2)]; and (X|£) is
negative, since the plane L is spatial. Hence {p \p) is positive, and p is
spatial.
This can be extended to any spatial subregion Pp by noticing that P^
has the property of a plane with respect to the region xP^.
(2) If the plane L and the point x be both anti>spatial, then the
perpendicular from d; to X is spatial, since it passes through the spatial
point |X.
(3) The line joining the poles |X and \U of two planes X and L' is
evidently the only common perpendicular to the two planes X and L\ It
is anti-spatial, if XX' be spatial : it is spatial, if XX' be anti-spatial.
428 HYPERBOLIC GEOMETRY. [CHAP. lY.
ForletZ=|Z and V ^\L\
Then if LL be spatial, by § 243 (4) {LL \LL') is positive. But
{lV\lV)^{LLLLy Hence «' is anti-spatial by § 243 (2).
If LL be anti-spatial, by § 243 (4) {LL \LU) is negative, and hence
{IV\IV) is negative. Therefore IV is spatial.
• (4) \i\LL = {\L\V) be spatial and L be spatial, then \LL' intersects L
in a spatial point. For let d be this point.
Then d = X|XZ'=(Z;|XO|X-(i|X)|Z'.
Hence (d | d) = (i | L) {LL \ LU),
But (Z [i) is negative, and {LL \LL) is positive. Therefore d is spatial.
The theorem also follows immediately from subsections (1) and (3).
263. Distance between Planes. (1) To prove that the distance
(hyperbolic or elliptic) between two planes is equal to the distance between
the feet of their commdn perpendicular line.
For let L and L be the two planes ; and let d = i \LL\ d! = L \LL,
Then d = {L\L)\L-{L\L)\L, d' ^{L\L)\L -•{L\L)\L
Hence {d\d')^{L\LJ-{L\L){L\L){L\L)
= ''{L\L){LL\LLy
Hence if LL be anti-spatial and {L \L) be positive, [cf. § 252 (3) and (4)
and § 242 (2)]
cosh^'= -(X|i')(Xi'|ZXr
7 ^{{L\L){L\L){LL\LLy]
{L\L) ,ZZ'
Hence dd' is the distance which has been defined as the measure of
separation between the planes.
(2) Secondly if LL be spatial, d and d' are anti-spatial and on an anti-
spatial line [cf. § 252 (3) and (4) and § 242 (1)].
(L \L^
Then cos Z dd' = /f/r i r! /r/i r/xi = cos Z LL,
V K^ \L) {L \L )\
Then Z dd' is the angle which has been defined as the measure of separa-
tion between the plane&
(3) Also [cf. § 211 (6)], when the distance formula can be applied to
two subregions Pf, and Qp, each of p — 1 dimensions, these subregions are
both contained in the same region of p dimensions ; and therefore they have
the properties of planea Hence they possess a single common perpendicular;
and, when Pf, and Qp are spatial and their common subregion anti-spatial, the
length of this (spatial) perpendicular is the measure of separation between
253, 254] DISTANCE BETWEEN PLANES. 429
the subregions ; also when the common subregion is spatial, the angular
length of this (anti-spatial) perpendicular is the angle between the sub-
regions.
264. Shortest Distances. (1) The least distance from a spatial
point 0? to a spatial plane L is the perpendicular distance xp, where p is the
foot of the perpendicular.
For let q be any other spatial point on the plane L, Then since [cf.
§ 251 (2)] the angle between px and pq is a right-angle,
cosh — = cosh — cosh — .
7 7 7
Hence xq > xp.
This length of the perpendicular will be called the distance of x from the
plane L,
(2) To find this distance ocp, write I for \L, then
p ^ xl\l =^ {x 1)1 — (l\l) X,
Hence (p \p) = (X |X) {xl \xl),
and {xp \xp) = {x \iy (xl \xl) = (xLy (xl \xl).
Thus sinh^- /-(^M ^ / (^^y - ±(^^)
7"V(^l^)(p|i>) V -(a^\a^){L\L)-y/{^{xx){L\L)y
This formula gives the distance from a spatial point ^ to a spatial
plane L.
(3) The greatest hyperbolic distance from an anti-spatial point x to an
anti-spatial plane L is the perpendicular distance xp, where p is the foot of
the perpendicular from x to i.
Let q be any other point on L such that xq is spatial : also xp is spatial
from § 252 (2). But pq is anti-spatial.
Hence cosh — = cos pq cosh ^ .
y _ " 7
Hence xq < xp:
(4) It follows from (1) of this article and from § 253 that the length of
the common perpendicular is the least distance between the spatial points of
spatial planes with an anti-spatial intersection and that this least distance is
what has been defined as the distance between the planes. The same holds
for any two subregions of the same dimensions with a single measure of
distance between them.
(5) A formula, analogous to the formula for Elliptic Geometry, in § 226
(1) and (2), can be found for the perpendicular distance of any point a from
any subregion Pp, of p - 1 dimensions.
480
HYPERBOLIC QEOMETBT.
[OHAP. IV.
Case I. If a and Pp be both spatial, then
S /-(aPplaPp)
sinh
7 V(a|a)
7 'V l.a|aKP|.|Pp)'
Case II, If a and Pf, be both anti-spatial, and aP^ be spatial
/-(g
7 V ^a|a)(Pp|Pp)-
Case //J. If o and P, be both anti-spatial, and aP, be also anti-spatial
sin 0 — ' '
~V(o.«:
^(i'pii'p)'
(6) To prove these fonnulaB first consider the distance of a from the
straight line F, Let 6 and c be two spatial points on F and let ap be the
perpendicular from a on P [fig. 1].
Then F=bc=pb.
Also by hypothesis (pa\bc)-0 = (pa \pb\ since pb = 6c.
Hence by formula (i) of § 216 (1)
(P\P) (iPa^ \P<^) = (P^ li>^) (pot I pa) - (pa Ipft)" = (pb\pb) (pa \pa).
Hence (P^\P^) « (pa6|jxi6) ^ (a6c|a6c)
(p \p) (a |a) (pt \pb) (a \a) (be \bc) (a \a) '
Now, if the hyperbolic formula hold,
.,8_ /-(palpg) /--(aF\aF)
y^W{p\p){a\arW(P\F)(a\ay
and if the elliptic formula hold
(pa|pa) _ / (aP|aP)
sinS
_ / {pa\pa) __ /Jf
V(l>|p)(a|a)"V(P
(p\p){a\a) ^/(F\F){a\ay
In Case II the hyperbolic formula holds; since P is anti-spatial, and
therefore the point, in which \F meets the two
dimensional region aF, must be spatial; re-
membering that the section of the absolute by
aF is real. But ap passes through this point.
These formulae may be extended to the general
case of subregions of /> — 1 dimensions by exactly
the same reasoning as that used for the analogous
theorem of Elliptic Geometry in § 226 (2).
Fio. 1.
266. Shortest Distances between Subregions. (1) Let P^ and Q.
be two non-intersecting sub-regions of the pth aud ath orders respectively, so
that /9 + o* < i;. A series of propositions concerning lines of maximum and
minimum distance between Pg^ and Q, can be proved analogous to those for
Elliptic Qeometry in § 227. Let it be assumed throughout this article that
256] SHORTEST DISTANCES BETWEEN SUBREOIONS. 481
p>a. Four different cases arise according as P^ and Q^ are respectively
spatial or anti-spatial. We will only consider here the single case in which
Pp and Qff are both spatial.
(2) It can be proved, as in § 227 (1), that a line (spatial or anti-spatial)
of maximum or minimum distance (hjrperbolic or angular) between them is
perpendicular to both.
(3) The polar regions \Pf, and |Q<, are both anti-spatial, and of the
{v^p)th and (i/ — <r)th orders respectively. In general the region \Qa
intersects Pp in an anti-spatial subregion of the (p — o-)th order at least. The
regions |Pp and Q^r do not in general intersect.
(4) Let g'l, Js, ... Jff be <r independent points in Q^, Then any point x
in Qa can be written %^q.
Also write
,,_ {(Sfo)Ppl(Sfe)Pp)
If X be spatial, and xp be the perpendicular to P from x, then [cf. § 252 (1)]
p is spatial. Hence by § 254 (5), Case I, sinh* — = — \*. If a?p be anti-
spatial, so that the elliptic measure of distance holds, sin' ^xp = \\
Hence in all cases of lines of maximum or minimum length between
Pf, and Q^, <r conditions of the type, k^ = 0, hold; where fi, fa, ... f^ are
successively put for f .
Thus by the same reasoning as in § 227 (3) a determinantal equation of
the 6th degree is found for V of the form
Oa - X« (9a \qj), O^-X" (j, Ig,), • • m ^aa " ^' (ft Ift),
where a^^ ^p^^^^ , a„- a,i — (p-|pj- ,
with similar equations defining the other o's.
(6) Hence there are in general a common perpendiculars to the two
subregions P^ and Q^, (p > a).
If Pf, and Q, had been interchanged in the above reasoning, so that a; is a
point in Pp, and
(a^kXQJQ-)'
482 HYPERBOLIC QEOMETRY. [CHAP. IV.
then an equation of the pth degree for X.' would have been found. But by
the formula (i) of § 226 (4)
(a^Qa\a:Q.) + (x\Q,lx\Q,)^(x\x)(Q,\Q;) (i).
Now by subsection (3) above x may be supposed to lie in the region
-Pp |Qir> which is a subregion of Pf^, Thus for all points x in this subregion,
^ I Q, = 0 ; and equation (i) becomes
(a^Qa\a^.) = (x\x){Q,\Q,) (ii).
Now differentiate fi, fa, ... fp with respect to any variable 0, and put x'
dp
for %-^p. Then equation (i) becomes, after differentiation,
{afQ,\xQ,)^'{af\Q,\.x\Q;)^{af\x){Q,\Q,),
But (^|Q^|.a?|Qa) = 0.
Hence (^U|^Q«r) = (^k)(Q.IQ.) (iii);
which holds for any point x in the subregion Pp 1 Qa, which has made any
infinitesimal variation to the position x + x'Sd in the region P^.
Thus differentiating \\ and using equations (ii) and (iii),
dd "^ (^k)HQ.|Q.)
Thus the infinite number of lines drawn from any point in PplQ^ to Q,y
which are not necessarily perpendicular to Pp, fulfil the conditions &t>m
which the equation of the pth degree is derived. The analysis of this
subsection could have been used for the corresponding subsection in Elliptic
Geometry [cf. 227 (4)].
(7) It follows by the method of § 227 (5) that the a feet in Q^ of these
a perpendiculars are the one common set of a- polar reciprocal points with
respect to the sections by Qc of the two quadrics (x\x) = 0, and (xPf, |a?Pp) = 0.
(8) It follows by the method of § 227 (6) that the <r common perpen-
diculars all intersect |Pp; and that the a points of intersection with | Pp are
mutually normal.
(9) It follows by the method of § 227 (7) that the a lines of the
perpendiculars are mutually normal ; and that therefore they intersect Pp in
a mutually normal points, which define a subregion P„^ of the <rth order.
(10) Also these theorems can be proved by the method of § 227 (8).
(11) One, and only one, of the a perpendiculars is spatial. For consider
a spatial point p in Pf, and a spatial point q in Q,. Then the distance pq
is real and finite; it varies continuously as p and q vary their positions
continuously on Pf, and Q^; and it approaches infinity as a limit, when p
or q or both approach the absolute.
256] SHORTEST . DISTANCES BETWEEN SUBREGIONS. 438
Hence there must be at least one position of pq, for which pq has a
minimum value. Thus there is at least one spatial common perpendicular to
Pp and Q^.
Let jP be a force on the line of this perpendicular: then by (9) the
remaining (r— 1 perpendiculars must lie in \F. Now |^ is anti-spatial
[cf § 289 (4)].
Hence the other (a — 1) perpendiculars are anti-spatial lines, and their
lengths must be measured in angular measure.
266. Bectangulab Rectilinear Figures*. (1) Let attention be
confined to rectilinear figures lying in a two-dimensional subregion. Then
the straight lines of the figures have the properties of planes in this contain-
ing region. Let the two-dimensional region cut the absolute in a real
section. Let all the rectilinear figures have all their comers spatial, unless
otherwise stated.
(2) A rectangular quadrilateral (a rectangle) cannot exist. For in such a
figure two opposite sides would have two common perpendiculars, contrary
to § 253 (8).
Fio. 2.
(3) Two alternate sides of any rectangular figure intersect on the
(anti-spatial) pole of the included side.
Thus [see fig. 2] let A, B, and C be three consecutive sides of a
rectangular figure, so that the angles at the intersections of A and B, and
of B and C, are right-angles. Let the closed conic in the figure be the
section of the absolute.
* These reenlts have not been given before, as liar as I am aware,
w. 28
434
HTPERBOLIC GEOMETRY.
[chap. IV.
Then [c£ § 251 (3)] A and C must intersect in h\ the pole of B with
respect to the absolute.
Hence, corresponding to the rectangular spatial figure formed by the
lines -4, By C, ..., there is the figure of which the anti-spatial comers
a', h\ c', ..., are the poles of the lines of the original figure. Let this be
called the reciprocal figure.
Then in the reciprocal figure each corner, such as 6', is normal to the two
adjacent comers, such as a' and c' : so that Q> \a) = 0 = (6' |c').
(4) Let the point of intersection of A and B be a, and the point of
intersection of B and C be 6. Then ah is the side of the given figure corre-
sponding to V, Also by § 246 (3), aV= ofc.
(5) A rectangular pentagon can be described as follows [cf. fig. 3] : take
any two mutually normal anti-spatial points, c' and d! ; and let a' be any third
Fig. 3.
anti-spatial point, such that aV and ddf are spatial. Let the line A be the
polar of a\ C of c', D of d' ; let B be the line aV and E the line a'd\ Then
the lines A,B,C, D, E, taken in this order, form a rectangular pentagon with
its comers spatial. Let A and G intersect in b', and A and D in e' \ then
a'h'c'de' is the reciprocal pentagon.
(6) The following formula holds for the rectangular pentagon, giving the
length of any side ae in terms of the two adjacent sides ah and de :
cosh — = coth — coth — (i).
7 7 7
Li order to prove this formula, assume that the two-dimensional region is
the complete region with respect to which supplements are taken ; so that
we may write
6' = |aV, e=\a'd\
266]
REOTANQULAR RECTILINEAR FIGURES.
435
Then, by subsection (4),
J J
cosh* — = cosh" — =
Q>'\ey
since
But
and
Hence
7 {V |6')(e' k')
(a'c'\a'dj
{a'c'\a'c')(a'd'\a'd')
(c'|d') = 0.
{a;\cy(a'\dy
(oV |a'c') ia'd' \a'd') '
a'cf
coth* — = coth* "^^ =
7
(a' \&y
y - (aV \a'c')
ooth«^
cosh* —
"^^^ T = -I'd'la'd')
coth«^coth«^.
(7) The reciprocal figure of a rectangular hexagon can be decomposed
into a pair of triangles conjugate with respect to the absolute. For, in
figure 4, let the conic be the absolute : take any three anti-spatial points
a\ Cy ef, so that the three sides of the triangle ac'e' are spatial. Let A, C, E
be the polars of these points ; and let B be the line aV, D the line ce\ F the
line e'a\
Then A, B, C, D, E, F, taken in this order, form a rectangular hexagon.
Let A and B intersect in a', B and (7 in 6, and so on. Thus abcdef is the
rectangular hexagon, and a!h'ddlf!f* is its reciprocal figure.
28—2
436 HYPERBOLIC GEOMETRT. [CHAP. IV.
(8) The formulae connecting the sides of a rectangular hexagon are
simply the formulae of § 248 (5) for a semi-spatial triangle.
For consider the semi-spatial triangle olce' : let a, y, 6 be the measoree
of the separation between its sides.
Then [cf. § 248 (5)].
cosh — = cosh — cosh smh — smh — cosh — ;
7 7 7 7 7 7
sinh - sinh -^ sinh —
and 2. JL ^.
smh — ' smh — smh —
7 7 7
But, by subsection (4), cd = ce\ ef = e'a\ ab = aV; and by § 253 (3),
a^afy y = bc, € = cfe.
Hence the formulae connecting the sides of the hexagon are
,cd ,a6 . ef , , ab . ^ ef ^^ af /..v
cosh — = cosh — cosh-^^^ — smh — smh -^^cosh-^ (u;;
7 7 7 7 7 7
. , afc . , cd • y sf
smh — smh — smh -^
and IL ±^ ± (iii).
, y de • 1 /ct . , 6c
smh — smh ^^— smh —
7 7 7
267. Parallel Lines. (1) Two spatial straight lines in a subregion
of two dimensions may intersect spatially, or non-spatially, or on the absolute.
In the first case let them be called secant*, in the second case non-
secant, in the third case parallel These parallel lines are not the analogues
of parallel lines in Elliptic Geometry, c£ § 234.
(2) If the straight lines be secant, then by starting from a spatial point
on either line the point of intersection can be reached after traversing a
finite distance. This case is illustrated in figure 5.
Fia. 5.
* Cf. LobatschewBky and J. Bolyai (2oc. cit.).
267]
PARALLEL LINES.
437
If the straight lines be non-secant, then the point of intersection has
neither a real linear nor a real angular distance from a spatial point on either
of the lines. This case is illustrated in figure 6.
Fio. 6.
If the straight lines be parallel then the point of intersection is at an
Pia. 7.
infinite distance from any spatial point on either of the lines. This case is
illustrated in figure 7.
(3) Let the two straight lines ac^ be intersect at a point o on the
absolute.
Then {ac\bc)^(a\b)(c\c)'{a\c)(b c) = -(a\c)(b\c).
Hence if d be the acute angle between ae and be,
006^=:
— (aclbc)
(a
o)(b\c) _^
cy(b\cy}
V [(ac I ac) (be \ be)} \/{(a
Hence tf = 0.
Therefore the angle, which two parallel lines make with each other, is
zero.
438
HTP£BBOLIO QEOMETRT.
[chap. IV.
(4) Any spatial straight line meets the absolute in two points O] and a,.
Hence through any spatial point p two
straight lines can be drawn in the
plane parallel to the given straight
line, namely the line po^ and the line
p(h.
From p draw the perpendicular pd
on to the line OiOs. The length betwe^
p and d is pd. Let the angle between
dp and pai or pa^ be called the 'angle
of parallelism ; ' there ia only one angle of parallelism, since it follows irom
the subsequent analysis that these angles are equal. Then the angle of
parallelism is a function of pd only. For in the right-angled triangle pdoi,
we have
Fio. 8.
But
Hence
cos Z Oi = sin Z p cosh — .
7
COSZOi^l.
sinZ» = sech — .
7
and
This relation can also be written in either of the forms,
cotZi} = sinh2_
y
where e is here the base of Napierian logarithms.
Let the angle of parallelism corresponding to a perpendicular distance, ^t
from the given straight line be denoted by H (J")*.
Then the formula above becomes
cot -^ = e-* .
It is to be noticed that when f=0, n(f)=i7r; as f increases, !!({;)
diminishes ; and when ^ is infinite, H (f ) is zero.
(5) It is possible to draw a straight
line parallel to two secant straight lines.
For let the two straight lines intersect
at an angle a; draw cp bisecting the angle
a; and produce it to p so that
^ = 7logcotg.
The perpendicular to cp through p is parallel
Fig. 9. to both the secant lines.
Cf. Lobatschewsky, loc. ctt.
258]
PARALLEL PLANES.
439
268. Parallel Planes. (1) Let the complete region be of three
dimensions, then the planes are ordinary two-dimensional planes, and the
subplanes are lines.
Let two planes L and L intersect in a spatial line LL\ and let this line
cut the absolute in the two points Oi and a,. Then through any two points
p and p in the planes L and U respectively two pairs of parallel lines can be
drawn, namely ^701,^01, and p<h>p'<h- Thus all the lines through Oi in the
two planes L and L' form one series of parallel lines distributed between the
two planes; and all the lines through a^ form another series. And both
series are parallel to the line of intersection of the planes.
(2) If the line LL' touches the quadric, the points a^ and a^ coincide,
and the two series of parallel lines coincide. The planes may then be called
parallel.
The condition that LL^ may touch the absolute quadric is
(ZZ'|ZZ') = 0.
Hence sinZZi'ssO, and z:ZZ' = 0. Therefore parallel planes are in-
clined to each other at a zero angle.
(3) The planes through any point p which are parallel to a given plane
L envelope a cone, which has p for vertex and the section of the absolute by
L for its section in the plane Z.
Let pd be drawn perpendicular to L, and let ah be any tangent line to
the absolute lying in the plane L and touching it at a.
Then the plane pab is one of the parallel planes through p. Let L' be
Fig. 10.
written for the plane pab. Now draw pn perpendicular to L\ Then pa, pd
and pn are co-planar. For pd and pn pass through \L and \L\ But the line
Z [Z' is the normal subregion to the line at. Hence |ZZ' passes through a,
440 HTPERBOLIC GEOIIETBY. [CHAP.IT.
since ah touches the absolute quadric. Hence the three lines pn, pd, pa are
co-planar.
Now the two lines da and pa are parallel. Hence the angle Z apd is
n (pd). But the angle Z apn = ^tt. Hence Z dpn = ^ II (pd). Thus if
through any point p, distance ^ from any plane L, all the parallel planes to Z
are drawn, the normals to these planes form a cone of which all the generatois
make an angle i^ — H (f) with the perpendicular from jp to X.
fib.
Hi-
CHAPTEK V.
Hypebbolic Geometby {continued).
269. The Sphere. (1) The equation of a sphere of radius p and of
spatial centre h is
{x\x){h\h)Qo&\i^^^{b\xf.
Every point of this sphere is spatial. For (6 |a?)' is positive and (6|6) is
positive ; hence {x\x) is positive [c£ § 240 (1)].
(2) The equation
^{x\x){h\h)^{h\xy
represents an anti-spatial locus, when h is anti-spatiaL For then {b\b) is
negative, and hence {x \x) is negative [c£ § 240 (3)].
(3) The equation
-'^(x\x){h\h)^{h\x^
represents an anti-spatial locus, when h is spatial For then (6 1&) is positive,
and hence {x \x) is negative.
But the equation represents a purely spatial locus, wheu h is anti-spatial.
For then (b\h) is negative, and hence {x\x) is positive. Let € be written
sinh - ; then [cf. 254 (2)] a is the distance of the spatial point x from the
spatial plane \h,
(4) Thus, if the equation
'n(x\x)Q)\h)=^{h\xy
be considered as the general form of equation of a sphere, there are two types
of real spatial spheres ; namely the type with spatial centre 6, of which the
equation is
(.W(H6,«o,h.J.(*l.)-;
and the type with anti-spatial centre b, of which the equation is
442 HYPERBOLIC QEOMETBT. [CHAP. Y.
(5) This latter surface is the locus of points at a given distance o- from
the spatial plane -B (= 1 6) ; and the equation can be written [cf. § 254 (2)].
-{x\x) (B \B) sinh« - = (xBy.
Let the spheres of this second type be also called Sur&ces of Equal
Distance ; and let the plane B be called the Central Plane.
(6) The spatial sphere is a closed surface. For firstly, let the centre be
spatial, and let it be taken as the origin, e, of a normal system of reference
points \nth. the notation of § 240.
Then (<r|;.) = |-g-...-^|. (e\a:) = ^(ele)^^.
Hence the equation of the sphere is,
^-^-...-^^ = sech«2^:
thatis, tanh»^^-^-...-^^=0.
Therefore the sphere is a closed surface [cf. § 82 (5)].
(7) Secondly, let the centre be the anti-spatial point, Ci, of this normal
system of reference points.
Then (ei\^) = ^i{ei\ed = -%
and -(ei|«i)sinh» - =
sinh* -
7 «!
Hence ^-^]- ... -^•^' = cosech«- ^.
Therefore the equation of the sphere becomes
<7
This is the equation of a closed surface [cf. § 82 (5)].
(8) It is to be noticed that this last closed surface touches the absolute
along the real locus of v — 3 dimensious given by the equations
(9) If the centre of a sphere be spatial, it lies within the surface.
For let 6 be the centre of
(x.\x) (b \b) cosh« ^-(b \xy = 0.
259] THE SPHERE. 443
Then if a? be any point, the point 6 + Xx lies on the surface, when
(6|6)»8inh»^+2X(6|a?)(6|6)8inh«^+X»((a:|a:)(6|6)cosh«£-(6|a;)>} = 0.
The roots of this quadratic for X are real, if
(6|.).(6W»„l..e-(H6)-K-W(»l*)0«h'e-(»l-)-!»po-tive;
that is, if (6 \wy cosh'^ — (x \x) (b \b) cosh* ^ is positive ;
that is, if — (bx \bx) is positive.
But [cf. § 243 (2)] since the line bx is spatial, this condition is ful611ed.
Hence [cf. § 82 (1)] 6 lies within the sphere.
Also if b be substituted in the expression
{x\x){b\b)Q08h^^-(b\xy,
there results (6 (6)*sinh"^ , which is positive.
Hence [cf. § 82 (9)] any point x, which makes this expression positive,
lies within the surface.
(10) But if the centre 6 be anti-spatial, then the equation of the sphere is
-(x\x)(b\b)Bmh^- "(blxy^O,
Then if x be any point, the point 6 + X^ lies on the surface, when
- (6 16)> C08h« - - 2 (a: 16) (6 16) cosh» - + X« {- (a? |a?) (6 16) sinh» - - (6 |a;)»} = 0.
ft# A^ ft/
The roots of this equation for X are real, if
(x \by (6 \by cosh" - - (6 16)« {(x \x) (b \b) sinh« - + (6 \xy] is positive ;
that is, if (a? |6)* — (a; | a?) (6 1 6) is positive ;
that is, if — (ajfe \xb) is positiva
But xb may or may not be spatial ; and therefore {xb \xb) may be positive
or negative.
Hence the anti-spatial centre of a sphere lies without the surface. But
it is interesting to note that any spatial line, drawn through the (anti-spatial)
centre, cuts the sphere in real points. Also substituting b in the expression
-(»|a?)(6|6)sinh»--(6|a:)«,
we obtain ^{b\by cosh' - , which is negative. Hence any point a?, which makes
this expression positive, lies within the surfiu^e [c£ § 82 (9)]. Thus any
444 HYPERBOLIC GEOMETRY. [CHAP.T.
spatial point on the central plane | b lies within the surface. For, if x be
such a point, (. |6) = 0. and - (. ].) (6 16) sinh' ^ is positive.
(11) It C5an be proved, exactly as in the case of elliptic Geometry [ct
§ 228 (3)], that the line, perpendicular to any plane and passing through its
pole with respect to a sphere, passes through the centre of the sphere.
Hence it follows as a corollary, that the perpendicular to a tangent plane
of a sphere through its point of contact passes through .the centre of the
sphere.
260. Intersection of Spheres. (1) The locus of the inteisectioii of
two spheres
fi^{x\x) = {c\x)\
lies on the two planes
Q>\x)^^{o\x)
These planes are respectively the absolute polar planes of the points
h c
But if 6 and c are both spatial and of standard sign, the point ^ +^
is spatial, and its polar plane is anti-spatial^ This plane can only meet the
spheres, which are entirely spatial, in imaginary points. If 6 and c are antf-
of*
spatial, one of the two points s T s must be anti-spatial, and hence its
Pi P2
polar plane spatial. The other point may or may not be anti-spatial.
These radical planes are perpendicular to the line be, since be passes
through their poles.
(2) It can be proved, as in the Elliptic Geometry [cf. § 228 (4)] that the
lengths of all tangent lines from any given point to a sphere with a spatial
centre are equal. Also if p be the radius, and b the centre of the sphere, and
r the length of the tangent line from x, then
eoshI= m .
^ cosh^^{(x\x)(b\b)}
Also, by an easy modification of the proof and by reference to § 246 (2),
we find when the centre 6 is anti-spatial, and the distance fix)m the plans
|6 is (7,
coshl^ <^
' aiv« n
amh-^/{-(x\x)(b\b)]
7
(
I
k
^
260] INTERSECTION OF SPHERES. 445
(3) The locus of poiDts, from which equal tangents are drawn to two
sr>- spheres with spatial centres b and c, and with radii pi and p^, is given by
Hi ^^ cosh ^ V(6 1 6) cosh ~ \/(c I c)
'o/i:
•f;-:
rc
This locus is the spatial radical plane.
^ ; * The cases when 6 or c, or both, are anti-spatial can easily be discussed.
(4) The theorems of § 228 (5) also hold, with necessary alterations.
(5) The angles of intersection of two spheres can be investigated by the
method of § 228 (8). Let the two spheres be
€{b\b)(x\x) = (b\xy.
and ^{c \c) (x \x) = (c \xy]
where e stands for cosh' — , if 6 be spatial, and for — sinh" — , if b be anti-
7 7
spatial [cf. § 259 (4)] ; and i; stands for cosh" ^ , or for — sinh" -^ , according as
c is spatial, or anti-spatial.
Then it can easily be proved, as in the analogous theorem of Elliptic
c Geometry, that the angles of intersection, to and <o\ of the two spheres, which
correspond to the two radical planes, are given by
i' ^ +V{6,(6|6)(c|c)}-(6|c)
'^'"-^{ie-l)iv-l)(b\b){c\c)\'
and «^'»-VK*-l)(i,-l)(6|6)(c|c)}-
Also let it be assumed that in all cases (b\c) ia positive. Four separate
cases now arise.
(6) Firstly, let b and c be both spatial.
cosh — cosh — — cosh —
Then cos to =
sinh - sinh ^
7 7
— cosh — cosh — — cosh —
and cos.fi)' = ' .
sinh^* sinh —
7 7
Since coth ^ coth - is necessarily greater than unity, (d' is always
y y
imaginary.
^
446 HYPERBOLIC GEOMETRY. [CHAP. V.
The spheres have one real intersection, if
— 1 < cos © < 1 ;
that is, if pi '^ p2<bc < pi + p^,
(7) Secondly, let b and c be both anti-spatial ; and let be be anti-spatial.
Let b = \B, and c = |(7; so that B and C are the central planes of the spheres.
Then, since be is anti-spatial, BC is spatial.
sinh — sinh — — cos < BC
Now cos 0) = f ,
cosh — cosh ~ j
7 7 I
— sinh ~ sinh ~ — cos < BG
and cos a>' = ' ■ .
cosh ^ cosh ^
7 7
Then oo and eo' are both necessarily real
(8) Thirdly, let b and c be both anti-spatial; and let be be spatial. Then
BC is anti-spatiaL
sinh — sinh — — cosh —
Then cos © =
cosh — cosh —
7 7
— sinh — sinh — — cosh —
7 7 7
0"i , <7,
cosh — cosh —
7 7
The angle o> is real, if
The angle (d' is real, if
BG < <ri — (Tj.
The first condition secures a real intersection on one radical plane ; the
second condition secures a real intersection on both radical planes.
(9) Fourthly, let b be spatial, and c be anti-spatiaL Let B be the dis-
tance from 6 to the central plane C.
Then [cf. § 254 (2)],
oinh^-f (ftg)
7 - ^h(b\b)(G\G)l/
cosh — sinh — — sinh -
Hence, cos <» = y y 7 .
sinh ^ cosh —
7 7
261] INTERSECTION OF SPHERES. 447
— cosh — fiinh ~ — sinh -
and CQB »' = 1 ^ 7 .
sinh — cosh —
7 7
Then « is real, if <ra — Pi < S < o"2 + Pi ; and co' is real, if 8 < pi — a^.
This condition for 6>' includes the conditions for to, since S has been
assumed to be positive.
(10) Now a spatial plane is a particular case of a sphere with an anti-
spatial centre c, when 0- == 0 ; the plane is then |c.
Hence from subsection (9) the plane L cuts the sphere, with spatial
centre 6, at an angle <» given by
. , Pi - (6i)
smh ^ cos 0) = -J. — /» i\wr \T\\ •
And from subsection (8), the plane L cuts the sphere, with anti*spatial
centre 6, at an angle q> given by
. <T, -jbL)
COSn — cos O) - /i/LiLx/r I r\\ •
7 'J[Q>\h){L\L)\
Hence putting <» == 0, the plane-equation of a sphere is
-(6|6)(Z|Z)sinh«2l = (6X)»,
when the centre is spatial ; and is
(6|«(Z|i)co.b.S=(»iy,
when the centre is anti-spatial.
261. Limit-Surfaces. (1) If 6 be on the absolute, the surface
denoted by
is called a limit-surface. It must be conceived as a sphere of infinite radius.
Since the centre is on the absolute, by § 259 (11) all the perpendiculars from
points on to their polars with respect to the surface are parallel lines.
(2) Let a distance h be measured from every point x on the above limit-
surface along the normal xh, either towards or away from h. Let y be
the point reached. Then x and X can be eliminated from
a: + X6 = y, cosh»^=,— f^f-^,, e^(x\x)^{h\xy.
The result, remembering that (6 16) = 0, is easily seen to be
■€»exp(^(y|y)-(6|yy|{6«exp(-^)(y|y)-(6|y)»| = 0.
448 HYPERBOLIC GEOMETRY. [CHAP. T.
The surface obtained by measuring towards b is therefore
7
The surface obtained by measuring from b is
7
Both these surfaces are again limit-surfaces with b as centre.
6'exp(-^)(y|y)-(6|y)' = 0.
by measuring from b is
=«exp(^)(y|y)-(6|y)» = 0.
(3) Now assume that the spatial origin e, of a normal system of unit
reference points e, ei, ... e^^i, is on the surface.
Let eei pass through the centre 6. Then 6 is of the form e±ei, say « +^.
The equation of the surface becomes
€'(x\x)=[x\(e-\-ei)]\
But since e is on the surface, we can put a?= 6 in this equation. Hence
€«=1.
The equation now is
(x\x)^[x\(e + e,)]\
This form, by its freedom from arbitrary constants, shows that all limit-
surfaces are merely repetitions of the same surface differently placed.
262. Great Circles on Spheres. (1) Let any two-dimensioDal
region, through the centre of a sphere aud cutting the sphere in real points,
be said to cut the sphere in a great circle. Accordingly a great circle is
in general defined by two points on a sphere, since these two points and the
centre of the sphere (if not coUinear) are sufficient to define the two-
dimensioual region. The radius of the circle is the radius of the sphere, and
the centre of the circle is the centre of the sphere. If the centre be anti-
spatial, the circle is the surface of equal distance in the two-dimensional
region from the line of intersection of the two-dimensional region with the
polar plane of the centre. The two-dimensional region, since it contains the
centre, is perpendicular to the polar plane of the centre, that is, to the central
plane of the sphere.
(2) If the centre b be spatial, and two points pq on the sur&ce define a
great circle, tJien the length of the arc pq of the great circle [cf. § 248 (4)]
is aysinh—, where p is the radius of the circle, and a is the acute angle
7
between pb and qb.
r
(3) Let the centre be anti-spatial. Consider any two points p and p'
on a surface of equal distance a from any given plane. Let the two perpen-
diculars from p and p' meet the given plane in q and ^. Then the length of
262]
GREAT CIBCLES ON SPHERES.
449
the arc jop', traced on the great circle joining p and p', can be found in terms
of q and q. For putting Jj' = 8, it is easy to prove that
cosh ^ = — sinh" - + cosh* - cosh - .
7 7 7 7
Hence when pp' and S are small,
27* 7 7\ 27*/
Therefore
pp' = 8 cosh - .
y
But ultimately, j5p' = arc pp\
__ - "arc pp m {T
Therefore , = cosh - .
«r 7
But if p" be any point on the arc pp' prolonged to a finite distance, and
'\ip"i(' be drawn perpendicular to the plane, it is obvious that
^^'^arc^'^^j^^
(4) Let d, be the centre of a spatial sphere of radius p, and let a, 6, c be
three points on the sphere. Let the acute angle between dh and c2c be oe',
that between dc and da be ff^ that between da, and d6 be y ; let the angle
between the two-dimensional regions dah and da/c be a, that between dab
and dhc be yff, that between dhc and dca be y. Then the three two-dimen-
sional regions can be conceived as planes in a three-dimensional region.
Hence by § 250 (2),
cos a' =s cos /S* cos 7' + sin ^ sin y cos a.
Now a, P, y are the angles of the curvilinear triangle formed by the
great circles joining a, 6, c. Also if &c, ca, a6 stand for the lengths of the
arcs of great circles, by (2) of the present article.
a' = — cosech ^ , fl' = — cosech - , 7' = — cosech - .
»y 7 7 7 7 '^'
W.
29
450 HYPERBOLIC GEOMETRY. [CHAP. V,
Hence
be ca ab ^ . ca , ab ^ .
cos — ^-» = cos ——— cos —— + sm — ^^— sin ^-^— cos a ,
7 sinh - 7 sinh - 7 sinh ~ 7 sinh ^ 7 sinh —
'7777 Y
with similar equations.
Thus the relations between the lengths of the arcs, forming- a triangle d
great circles on a sphere of spatial centre, and the angles betiveen them are
the same as the relations between the sides and angles of a triang^Ie in an
Elliptic Space, of which the space constant is 7 sinh - . Thus an Slliptie
Space of I' — 2 dimensions can always be conceived as a sphere of radiiw
p with spatial centre in Hyperbolic Space of »^ — 1 dimensions, the great
circles being the straight lines of the Elliptic Space, 7 being the space
constant of the Hyperbolic Space, and 7 sinh - that of the Elliptic Space.
(5) Let a sphere with anti-spatial centre be a surface of equal distance <r
from a spatial plane; and let a, 6, c be three points on the sphere, and a\ h\ c'
be the feet of the perpendiculars from a, 6, c on to the plane of equi-distanca
Let a, 6, c be joined by great circles of lengths 6c, ca^ ah.
Let a, /3, y be the angles of the curvilinear triangle ahc ; they are also
the angles of the triangle a'Vc\ since the two-dimensional regions containing
the great circles are perpendicular to the plane of equal distance.
Then [cf. § 248 (2)]
6/ t f 'f JTj I I 777
^ c , ca , ao . , ca . , ao
COSH — = cosh — cosh sinh — smh — cos a.
7 7 7 '^ '^ ^
n
But by subsection (3) of the present article, h'd = — , with similar
cosh-
7
equations. Hence
16c ^ oa .ah . ^ ca . , ah
cosh — — = cosh — cosh — — ^— — sinh -^^ ^^ smh — — — ooso.
7 cosh - 7 cosh — 7 cosh - 7 cosh — 7 cosh —
7 7 7 7 7
Thus the relations between the lengths of the arcs, forming a triangle of
great circles on a sphere of equal distance a- from a spatial plane, and the
angles between them are the same as the relations between the sides and
angles of a triangle in a Hyperbolic Space, of which the space constant is
7 cosh - . Thus, since 7 cosh - is always greater than 7, a Hyperbolic Space
of 1/ — 2 dimensions can always be conceived as a spherical locus with anti-
spatial centre in a Hyperbolic Space of 1/ — 1 dimensions and of smaller
space constant.
263] GBEAT CIRCLES ON SPHERES. 451
(6) The relations between the sides and angles of a curvilinear triangle
formed by great circles on a Limit-surface can be found either from (4) or (5)
by making p or a- ultimately infinite. Then with the notation of (4) or (5)
6c" = ca" + a6" — 2ca.a6co8a.
Hence triangles formed by great circles on Limit-surfaxses have the same
geometry as triangles in ordinary Euclidean Space; for instance, the sum
of the angles of any such triangle must equal two right-angles. Thus a
Euclidean Space of i^ — 2 dimensions can be conceived as a Limit-surface in a
Hyperbolic Space of j^— 1 dimensions*.
ft
263. Surfaces of Equal Distance from Subregions. (1) Let P^,
be a spatial subregion of p — 1 dimensions, and let Pp be the regional element
of the pth order which represents it. Then locus of points oc at the given
distance B from this subregion is by § 254 (5),
(x\x) {Pp |Pp)sinh«- +(a?PpkPp) = 0.
7
(2) Now take as reference elements v normal points, of which the spatial
origin e and p — 1 other points ei, e^, ... e^i lie in P^, and the remaining
p — p elements lie in |Pp. Also let e be at unit spatial intensity, and
^, «a, ..• «r-i at unit anti-spatial intensity. Let Pp = e^ ... Cp-i. Then
(Pp|Pp)=(-l)p-^ Let
a? = fe -h fi6i + ... + fp_i«p-i + Vp^p + ... + ^,.-16^-1.
Then (xP^ \xPp) = (-!>» tr^, (a? k) = f* - f i« - . . . - 17V1.
Hence the equation of the surface of equal distance from Pp becomes
that is,
(P-fi'-...-r^i-V-..--^Vi)8inh»^ = 2^»;
(p-fi»-...-p^i)tanh>--(V + ...+i;Vi) = 0.
7
This is a closed surface with no real generating regions. Hence the
parallel regions of Elliptic Space [cf. § 229] have no existence in Hyperbolic
Space.
* The idea of a epaoe of one tjpe as a locus in a space of another type, and of dimensions
higher by one, is dne partly to J. Bolyai, and partly to Beltrami. Bolyai points out that the
relations between lines formed by great i^cles on a two-dimensional limit-sorfiace are the same
as those of straight lines in a Euclidean ^lane of two dimensions. Beltrami proves, by the use
of the pseudosphere, that a Hyperbolic space of any number of dimensions can be considered
as a locus in Euclidean space of higher dimensions. There is an error, popular even among
mathematicians misled by a useful technical phraseology, that Euclidean space is in a special
sense flat, and that this flatness is exemplified by the possibility of an Euclidean space containing
surfaces with the properties of Hyperbolic and Elliptic spaces. But the text shows that this
relation of Hyperbolic to Euclidean space can be inverted. Thus no theory of the flatness of
Euclidean space can be founded on it.
29—2
452
HYPERBOLIC GEOMETRY.
[chap. V.
264. Intensities of Forces. (1) Consider an extensive manifold
of three dimensions. The only regional elements are planar elements and
forces. A spatial planar element X is at unit intensity when (X|X) = — 1,
and an anti-spatial planar element X is at unit intensity when {X\X)^\
[cf. § 240 (3)].
(2) In order to determine the intensity of a force xy, let it be defined
that the intensity of o^ is some function of the distance ^, or ^ ^ if the
measure of distance be elliptic, multiplied by the product of the intensities of
X and y. Then by the same reasoning as in § 230 (2) for Elliptic Space it
can be proved that : (a) if x and y be both spatial, the intensity oi xy \s
fsj[{x \x){y \y)] sinh — , that is {—xy\xy]k\ where it is to be noticed that, by
7
§ 243 (2), {xy \xy) is negative, when xy is spatial : (/9) if a; and y be both anti-
spatial and xy be spatial, the same law of intensity holds as in (a) : (7) if xy
be anti-spatial, the intensity of xy is \/{(^l^)(y ly)} sin^, that is (xy\xy>^.
Hence the intensity of a spatial force F is ['-F\F]^, that of an anti-spatial
force i^ is {i^ \F]K
(3) If P and Q be two planes the standard form of a real force of the
t3T>e xy is iPQ, If the force be anti-spatial, its intensity is
VKP|P)(Q|0)}8inh^;
7
if the force be spatial, its intensity is \/{{P \P)(Q\Q)] sin i^Q.
266. Relations between two Spatial Forces. (1) In general [cf.
§§ 231 (1) and 255 (6)] there are only* two lines intersecting the four lines
d r b
Fio. 2.
F, F\ \Fy \F\ Let these be the lines ah and cd. Then ah and cd are
perpendicular to both F and F' ; also each is the polar line of the other.
One of the two must be spatial and the other anti-spatial. Assume ah
spatial. Let ab^hy and jLcd—Q. Then S i3 the shortest distance between
the lines, and 0 will be called the angle between the lines.
* For the diBoassioii of an ezoeptional case for imaginary lines see the oorreaponding dieeas-
sion for Elliptic Space, cf. § 234, in which case the lines are real.
264, 265] RELATIONS BETWEEN TWO SPATIAL FOBCE8. 453
Let F^ao, F' ='bd.
Then cooh ^ = <'' l^> co8^ = -^^^M__
also (^|^) = (ac|M)=(a|6)(c|d), since (a|d) = 0 = (6|c).
Also (F\F) = (ac\ac) = (a\a)(c\c), since (o|c) = 0.
And (F' \F') = (6 16) (d \d), since (6 |d) = 0.
Henc« (^l-FQ _ (a|6)(c|d) ..agcoah^
^^°*^ VK^I^X^'I^)} - V{(a |a) (6 1 b) (c |c) (d |d)} -^f''^^^-
(2) Again, let a' be the point normal to a on the line ab, and c' be the
point normal to c on the line cd. Also let (a'|a') = — (a|a), where a is
assumed to be spatial, and (c' \c)=(c \c).
, a cosh - + a' sinh - , /> . / • /»
7(b\b) 7(a\a) ' ^^ V-(d|d)" V-(c|c) '
Hence
./EfEf/x •/ Lj\ i (oa'ccO sin ^ sinh -
%{FF^ _ I (acta) «___ 2
VpT^T(F|F)} ~ ^[{a\a){h\h)(p\c){d\dj] -(a|a)(6|6)
= ± sin tf sinh - ;
7
since [cf. § 240 (6)] i (oa'cc') =* ± (a | a) (6 1 6).
(3) If the forces intersect, the point of intersection is either spatial or
anti-spatial or on the absolute. If the point of intersection be spatial, then
S = 0. Hence {FF') = 0 ; and
^[{F\F){r\F)] ^^'^•
If the point of intersection be anti-spatial, then ^ = 0. Hence {FF') = 0 ;
and
^{iF\F){F' \F')] '^''y-
If the point of intersection be on the absolute, so that the lines are
parallel, then S = 0 = ft Hence (FF') = 0, and
(F\Fy^(F\F){F'\F').
(4) Let two forces F and F' have a spatial intersection, and let their
intensities be p and p'. Let the single force F+F' he of intensity c. Let 0
be the angle between F and F\ Then
a^=:^(F + r)\(F'hF')=:p^'hp'^±2ppcoB0.
The upper sign of the ambiguity must be chosen so that, when 0^0,
a' = p + p\
464 HYPERBOLIC GEOMETRY. [GHAP. V.
If F and F' be spatial but have an anti-spatial point of intersection, let £
be the shortest distance between F and F', Then as before, if -P+ jP' be
spatial,
o^ = p^ + p'« + 2,pp' cosh - .
But it is possible that, though F and F' are spatial, F+F' may be anti-
spatial, the intersection of F and F' being anti-spatial. In such a case
o-> = 2pp cosh - - p» - p'l
266. Central axis of a System of Forces. (1) It has been proved
in § 175 (14), also cf. § 232, that any system of forces has in general one and
only one pair of conjugate lines, which are reciprocally polar with respect to
a given quadric. Now let a system 8 have the two conjugate lines diO^, a/14,
which are reciprocally polar with respect to the absolute. One of the two must
be spatial, the other anti-spatial. Let OiO, be spatial. Then 8='Kaia^ + fia^^;
and this form of reduction is unique. The line OiO^ will be called the central
axis of the system. Let the points Oi, a,, a,, 04 be so chosen at unit intensity,
that
8 = OiOa + 0^4 ;
then Old, ( = S) and Z 0^4 (= a) will be called the parameters of the system.
A system, 8, referred to its central axis may also be written in the form
OiOa-hwIaiOs,
where € is real.
Then (88) = 2i€ (aiO^ ]aia^), and (fif |iS) = (1 - €«) (a^a^ \ OiO,).
(2) Let 8 denote the system F-hielF, and 8' the system jP' + ii; \F\
Also with the notation of § 265, let S be the shortest distance between the
lines and 0 the angle between the two.
Then
(S8r}^(l^€v)(FF') + i(€ + v){F\F')
= {{F\F)(F' \F')]i |± i(l - €17) sin tf sinh - + i(€ + i7)co8^co8h -I .
And
{8m = i(e + v)iFF')-^(l^ev)iF\F')
= {{F\F) (F' \r)}i \±(€ + v) sin 0 sinh - -h (1 - €17) cos ^ cosh -I .
(3) The simultaneous equations (88^) = 0, (8\8')^0, secure that the
axes of 8 and 8^ intersect at right angles.
For from (2), unless e or 1; be i, which is the case of an imaginary system
analogous to the real vector system of Elliptic Geometry [c£ § 236], (SS') = 0
and (S\8') = 0 entail cos 0 cosh - = 0, sin tf sinh - = 0 ; that is S = 0, ^ = s •
' 7 7 2
266, 267] CENTRAL AXIS OF A SYSTEM OF FORCES. 456
(4) Every dual groap contains one pair of systems, and only one pair,
such that their axes intersect at right angles. The proof is exactly the same
as for the analogous theorem of Elliptic Qeometry, cf. § 232 (4). Let this
pair of systems be called the central systems of the group, and let the point
in which their central axes intersect be the centre of the group.
(5) Dual groups with real director lines can be discriminated into three
types according as, either (a) both director lines are spatial, or (fi) both
director lines are anti-spatial, or (7) one director line is spatial and one is
anti-spatial.
(6) To find the locus of the central axes of a dual group, let e be the
centre, eCi and ee^ the axes of the central system, and ee^ a line perpendicular
to the lines eei, ee^. Also let eete^ be a normal system at unit intensities.
Let Si^€ei + i€i\eeiy iS, = ees + 1«2{^^» be the central systems of the group*
Now [cf. § 240 (5)] we may assune
* I * I I * I *
Any other system S' of the group can be written
iS' = XiSi -h \S% = « (^«i + ^A) + i I . e (^l€lel -h Xae^ea).
Then, as in § 232 (5), this system can be identified with the system
(« + K^) (jhfii + 11^ + ie I . (e + C«,) (ji^ei + yM,).
Hence all the central axes of systems of the group intersect ee^ at right-
angles. Let eei be called the axis of the group.
The equation to find € is
(6» - 1) {6,«X,« -h 6,«X,»} - e {(€,«- 1) X,« +(€.'- 1) VI = 0.
The locus of any point 2f e on an axis of any system of the group is
(61 - 6.) U, (p - f,«) = (1 + ^,e,) f f 3 (f i» + f ,').
267. NoN-AxAL Systems of Forces. (1) A system of forces, not
self-supplementary, and such that {8S) = ±(S ]S), is called a non-axal system
[cf. § 233].
(2) All such systems are imaginary. For if S be real, then [cf. § 240 (5)]
it is easily proved that (88) is a pure imaginary, and that (£f |/S) is real.
(3) Hence, firom this article and from § 266 (3), any system 8, such that
(88) = ± (S |jS), whether it be self-supplementary or not, is imaginary.
(4) Accordingly the theorems of § 266 hold for all real systems of forces.
CHAPTER VI.
Kinematics in Three Dimensions.
268. Congruent Transformations*. (1) A congruent transformation
is a linear transformation, such that (a) the internal measure relations of
any figure are unaltered by the transformation; e.g. if abc is transformed
into a'6V, then ab = a'b\ and the angle between ab and ac is equal to
that between a'b' and ac', and similarly for the other sides and angles:
(fi) the transformation can be conceived as the result of another congruent
transformation p times repeated, where p is any integer: (7) real points
are transformed into real points: and (8) the intensities of points are un-
changed by transformation.
It follows from (a) that points on the absolute must be transformed into
points on the absolute.
Hence congruent transformations must transform the absolute into
itself.
It follows from (a), (13) and (7) that spatial points must be transformed
into spatial points. For from (a) [cf. § 241 (1)], either all spatial points are
transformed into spatial points, or all spatial points into anti-spatial point&
Also from (13), since the integer p may be taken indefinitely large, a finite
transformation can be considered as the result of p repetitions of an infini-
tesimal transformation. But an infinitesimal transformation must transform
spatial points into spatial points. Hence the same holds for a finite trans-
formation.
(2) Let the discussion be now confined to regions of three dimensions.
To prove that a congruent transformation must transform the absolute by
a direct transformation (c£ § 194).
For by (13) of (1) any congruent transformation can be conceived as
the result of p repetitions of another congruent transformation. But an
even number of applications of either a direct or a skew transformation of a
quadric produces a direct transformation of the quadria Hence every
congruent transformation is a direct transformation of the absolute.
* The theory of congruent transformations is due to Klein, of. loc. cit. p. S69 ; Bnohheim has
applied Grassman's algebra to this subject, cf. loc. cit. p. 370.
268] CONGRUENT TBANSFORMATIONS. 467
(3) By § 195 (2) to (6) among the latent points of a direct traDsforma-
tion of the general type there are the points of intersection of two conjugate
polar lines with the quadric. Now in Elliptic Space the absolute is imaginary;
and therefore the co-ordinates of the latent points on either one of the polar
lines, referred to real reference points, foim pairs of conjugate imaginaries.
In Hyperbolic Space one polar line must be anti-spatial and one is spatial:
the co-ordinates of the latent points on the anti-spatial polar line, referred
to real reference points, are pairs of conjugate imaginaries : the latent points
on the spatial polar line are two real points, and their co-ordinates are real.
Now either in Elliptic or in Hyperbolic Space let Oi, a,, a,, 04 be the
four above-mentioned latent points of a congruent transformation, and let
OiOa and 0^4 be conjugate polar lines. Also let Og and a^ be imaginary
points, then their co-ordinates are pairs of conjugate imaginaries. Hence
if a, and a^ be taken as reference points, the co-ordinates, rj^ and 1/4, of a real
point y (= 17^ + 1/404) are conjugate imaginaries.
(4) Let the latent roots of the congruent matrix be tti, a^, 0^9, ou^ Then y
is changed into v^'^^^ + V^'^i'^if *^d ^Y (7) of (l)i V^/h + Vi^i'^i is a real point.
Hence ec, and 04 must be conjugate imaginaries.
Similarly if Oi and o^ are imaginary points as in Elliptic Space, a^ and 0^
are conjugate imaginaries ; but if Oi and Og are real points as in Hyperbolic
Space, tti and Os must be real.
Hence in Elliptic Space the latent roots are two pairs of conjugate
imaginaries, in Hyperbolic Space one pair are real and one pair are conjugate
imaginariea
(5) In Hyperbolic Space both the real latent roots ai and Os must
be positive. For the given congruent matrix may be conceived, according
to (J3) of subsection (1), as the result of another congruent matrix twice
applied. Let i8i, A, A> A be the latent roots of this matrix. Then )8i" = ai,
and /Ss' = Og. But fii and yS, are real by the same proof as that for a^ and a, ;
hence ai and cr, are positive. Therefore the real roots of a congruent matrix
in Hyperbolic Space are positive.
(6) By § 195, aiOa = 0,^4. Hence in Elliptic Space we may put,
And in Hyperbolic Space we may put,
a a
Also by (S) of (1) the intensity of any point is unaltered by trans-
formation. Now the intensity of i/iOi -h i/jO, is {rjirf^ (oi loa)}* and the
intensity of the transformed point is
458 KINEMATICS IN THREE DIMENSIONS. [CUAP. YI.
Hence X » 1. Thus the latent roots in Elliptic Space take the form
01 = 6^, aa = e y, a^ — ey, a^^e y;
and in Hyperbolic Space
ai = ey, a^ — e y, aj = 6^, (u = er\
(7) The special type of direct transformation, with only three semi-
latent lines [c£ § 195 (7)], cannot apply to Elliptic or Hyperbolic Space, so
as to give a real congruent transformation. For, with the notation of § 195 (7),
the points «i, 62, e^, 64, and the planes eje^t and ^1^04 are imaginaiy, both in
Elliptic and Hyperbolic Space. But an imaginary plane always contains one
straight line of real points. Hence the semi-latent plane eie^ contains one
real line. But, since real points are transformed into real points, this line
must be semi-latent. Also the semi-latent lines eie^ and eiCs, which lie in
this plane must be imaginary, since they are generators. Hence a third
semi-latent line must lie in this plane ; and this is impossible in this type
of transformation.
(8) The theory of congruent transformations in Hyperbolic Space will
first be discussed, cf. §§ 269 to 280, and then that of congruent transforma-
tions in Elliptic Space, cf. §§ 281 to 286.
269. Elementary Formulae. (1) Let Oi, a,, a,, a^ be the latent
points of a congruent transformation in Hyperbolic Space.
Let OiOa and 0,04 be conjugate polar lines ; and let them be called the
axes of the transformation. Let OiOs be spatial, and be called the spatial
axis, or more shortly the axis ; then a^^ is anti-spatial, and may be called
the anti-spatial axis. Thus Oi and a, are real ; and a, and a^ are imaginary.
Then it at once follows that,
(a, ai) = 0 = (0, la,) = (a,|a,) = (a4 \a^) = (ai \(h) = ((h Ia4) = (a2|a,) = (a, |a4).
(2) Let 6, ei, 69, ^ be a normal system of elements at unit intensity,
of which e is the spatial origin.
Let eei be the line OiO,, and e^ the line dya^.
Then by § 247, we may write
ai = ei + e, a, = ei--6, a^^e * e^ + e* e^, a^^e^e^ + e * e^.
Hence (oh joa) = (^i |ci) — (e |e) = — 2 ;
and (a, ^4) = (e, {e^) + (e, |e,) = - 2.
Also from § 240 (5) (eeiC^) = %.
Hence {(x^d^ot^^) = — 4i (eeie^) = 4 = (Oi | a,) (a, | ai).
269, 270] ELEMENTARY FORMULAE. 459
(8) Again, by substituting for Oi, a,, a,, a* in the expression for any real
point X (= f itti + f A + f ao, + f ^a*), we find
But since x is real, the coefficients of 6, ei, ^, ^ are real. Hence f, and
^4 are conjugate imaginariea Let f , = pe^^ (^ = pc"^.
Then aJ = (fi-fO<^ + (fi+f2)ei + 2pcos(tf-^)e, + 2pcos^^ + j)e,
= i;e + i7iei + 17268 + 17A (say).
Any spatial point x must satisfy the condition that (x\x)he positive.
But {x\x)=^ 2fifa (oi |a,) + 2f,f,(a3 |a,)
= -4(f,f, + f,f,) = -4(f,f, + ^«).
Hence for a spatial point f if a is negative, and — f if , > />'.
(4) The congruent matrix transforms x into x\ where
a a
= (fiey-fa<5 >) e + (fieri' + fa<5 9)e,
+ 2p cos ^tf + a -^j e, + 2p COS ^^ + a + 1^)^
= 1 11 cosh - + Wi sinh -]e + lfi sinh - + t/i cosh -J e,
\ 7 7/ V 7 7/
+ («79 cos a + 17, sin a) ea + (i/s cos a — 17a sin a)6,.
270. Simple Geometrical Properties. (1) Consider any point x
a _a
(=fifli + fA) on the axis OiO^. It is transformed into a/ = f iC^Oi + f a^'^Oa,
which is again on the axis.
Furthermore,
a^
^ = |logK,aiaa) = |log-^^»=|log6^ = 8.
Thus all points on the axis are transferred through the same distance B.
(2) Again, consider any plane P through the axis OiO, and any point
X (= fs^ + ^^4) on the anti-spatial axis a^^. Let the plane OiO^ be called
Agy the plane OiO^i be called A^. Then A^ and A4 are the two planes
through OiOa which touch the absolute in imaginary points a, and 04. Also
P = f»4, + ^4^4; and P is transformed into the plane P' = f«e**-4, + ^^bt^A^.
Furthermore, ^ PP' = ^ log (PP', ^,^14) = ^i ^""^ ffl'.^, = «'
Hence every plane through the axis is rotated through the same angle a.
460 KINEMATICS IN THREE DIMENSIONS. [CHAP. VI.
(3) The distance, «r, of any spatial point x from the axis is given by
sinh - = f {<c<h(h\<c<h(id I* ^ _2p .
7 [- {x \x) (oicu, loiOa) J (x \x)^ '
where x is written in the form f lO^ + faOa + pe^a^ + pe^^a^. But the trans-
i -i
formed point a/ is fiC^ai + fae Ya3 + pe*(*+*>a3 + pe~*(*+»>a4; and it is obvious
that its distance from the axis is the same as that of x. Hence the distance
of a point from the axis is unaltered by the congruent transformation. Also
it is easy to prove that
cosh g = \:^\^ = {- <^- '^->l*
(4) To find the distance of the transformed point x' from the plane
through X perpendicular to the axis OiO, of the transformation.
This plane is represented by xa^^. Now xa^^ = 2ixe>^] but 2ixe^ is in
the standard form of §240 (2); hence xa^^ is in the standard fonn. Now
if ^ be the distance of a/ from this plane, it is easily seen after some
reduction that
sinh - = + -77 — , , I .. ,——, vT = cosh — sinh - .
7 ^j [— {x \af) {xa^^\xa^^] y y
Hence ^ is independent of a, and depends only on S and on the distance of
X from OiOa. Also when 8 = 0, f=0; and when -btssO, f becomes S in
accordance with subsection (1).
(5) It is easily proved that
— / / I /\ fif,co8h- + p*cosa ^
cosh — = ,,/ I w /|— /vT = iTir^ — 9 =cosh'— cosh- + smh*— cosa
7 V{(«? k) (^ F )} fif2 + p 7 7 7
271. Translations and Rotations. (1) Let the quantities S and a
be called the parameters of the transformation.
If a = 0, the congruent transformation is called a translation through the
distance S with aiO^ as axis. The effect of the translation on a point x is
that the transformed point Xi is at the same distance as x from Oia^, is in the
( ^'\
plane xOnO^t £>^d is at a distance 7sinh~MC0sh ~ sinh -{■ from the plane xcL^i,
( 7 7j
where isr is the distance of x from OiO,, or in other words the plane Xia^4 is
at a distance B from the plane xcl^^, OiO^ being their common perpendicular.
(2) If S = 0, the congruent transformation is called a rotation through
the angle a with Oja^ as axis. The effect of the rotation on the point x is
that the transformed point x^ is at the same distance «r from OiO^, and is
in the plane osa^i, and that the plane x^a^ makes an angle a with the
plane xUiOq.
?
271]
TRANSLATIONS AND ROTATIONS.
461
(3) It is obvious that the general congruent transformation, axis aiO^,
parameters (8, a), in its eifect on any point x is identical with the effect first
of the translation, axis aiO^, parameters (8, 0), bringing ^ to o^i, and then of
the rotation, axis OiO,, parameters (0, a), bringing Xi to x'; or first of the
rotation bringing a; to ^, and then of the translation bringing x^ to x\
It is to be noticed that congruent transformations with the same axis
are convertible as to order, but that when the axes of the transformations
are different the order of operation affects the result.
It will be convenient for the future to use the letter K for the matrix
representing a congruent transformation, so that Kx is the transformed
position of x,
(4) A further peculiarity of translations and rotations is established
by seeking the condition that a real plane, other than one of the faces of the
tetrahedron OiO^fl^i may remain unchanged in position, i.e. be semi-latent.
Let X be any point in such a plane. Then x, Kx, K^x, K*x must all lie in the
plane. Therefore (xKxK^xK^x) = 0.
But
a _a
88 as
88
38
K*x = f i«> a, + f ^ y(h + p€^ <•+**> (h + pe-' <*+»»)a4.
Therefore
{xKxK^xE}x) ^ ^ ,
1,
1 .
1 ,
1
t
e'y.
«+<•,
e-^
it
ey,
e'y,
e+«",
e-«-
»
ey.
e'y.
e+«",
g-Sfa
Now f 1 = 0, or f a = 0, or p = 0, each makes the plane to be one of the three
real £aces of the tetrahedron (OiO^a^^, Hence the determinant must vanish,
if another real plane is semi-latent. But this condition can be written.
The only solutions, making 8 and a real, of this equation are S =- 0 or a = 0.
Thus rotations and translations are the only congruent transformations
by which planes do not change their positions. For rotations, planes per-
pendicular to the axis are rotated so as to remain coincident with them-
selves; for translations, planes containing the axis are translated so as to
remain coincident with themselves. In other words, the only possible motions
462 KINEMATICS IN THREE DIMENSIONS. [CHAP. YI.
of a plane, which remains coincident with itself, are either rotations about a
perpendicular axis, or translations along an axis lying in it.
Similarly it is obvious that rotations and translations are the only con-
gruent transformations by which points — other than the four comers of the
self-corresponding tetrahedron — do not alter their positions. For rotations
these points are points on the axis, for translations they are points on the
anti-spatial axis.
(5) The property, which discriminates a translation from a rotation, is
that, as a translation is continually repeated, the distance between the original
and final positions of any spatial point grows continually greater. For let
the translation, axis OiO,, parameters (8, 0), be repeated v times, and let
Then K^x = ^^ey Oi + ^^y (h + pe^a» + pe'*^ a^.
Hence (x {K^'x) = — 4f ifj cosh 4p'.
- f 4f ,?, cosh - + 4p«')
.-1^ y /
Therefore xK^'x =^ y cosh' — , , ^ — .
(x\x)
Hence xK^x grows continually greater as i/ is increased. But in the case
of a rotation, if the parameter a bear a commensurable ratio to four right
angles, then after a certain number of repetitions every point coincides with
its original position.
272. Locus OF Points of Equal Displacement. (1) The locus of
points, for which the distance of displacement [cf. § 270 (5)] in the general
transformation, parameters (8, a), is equal to a given length a, is the quadric
surfiu^e
2f if 2 cosh - + 2p« cos a = cosh - {2f if , + 2p^\ ;
that is (x \x) f cosh cos a j = — (cosh — cos a] (x joi) (x |a,).
Also by referring to § 195 (2) we find that all quadrics of the form,
(x\x)-^ /*» (x |oi) (a? la.) = 0,
remain coincident with themselves after the congruent transformation, axis
OiO,. They are quadrics which touch the absolute at the ends of the axis
OiOa. But by comparison with the quadric, which is the locus of points
transferred through a given distance <r, we see that the system of quadrics
found by varying a- is the same* as that found by varjring fju
• Of. Sir R. S. BaU, «* On the Theory of Content," Transaetions R. I. A,, loe, cU. p. 870.
272, 273] LOCUS of points of equal displacement. 463
In fact we have
(cosh cos a]
/*" = 7 — J T-
(cosh cosa)
(2) The sections of these quadrics by planes perpendicular to aiO, are
circles. For let pa^^ be any such plane, and let a? = fp + ^^ + ^4a^> Then
(«|ai)-?(i>|ai), (x\a^)=^(p\<h); and (ica^J>)^^(pa^J)), where 6 is any
fixed point.
Hence (a;\a,)(x\a,)^^^-^^^ (axi^Jby.
Therefore the section of the quadric,
(x \x) + /*« (a? Oi) (a? lojj) = 0,
by the plane porfl^ is the intersection of the plane with the sphere
ipa^Jb)
(3) The centre of this sphere is the point 1 0,046, that is the point where
OiO, meets \b. Hence the centre of the circle, which is the intersection of
pa^4 with the quadric [cf. § 260 (4)], is the point where OiO, meets
p(i^4. It is otherwise evident from § 270 (5), that all points on such a
circle must receive equal displacements, and that the distances of their
displaced positions from the plane of this circle must be the same for each
point. Similarly the curves of intersection of the system of quadrics with
planes through OiO,, that is of the form jmho,, are lines of equd distance
from OiOs, that is are circles with their anti-spatial centres on 0^4.
273. Equivalent Sets of Congruent Transformations. (1) Any
congruent transformation (K) may be replaced by a combination of two
transformations consisting first of a translation with its axis through any
arbitrarily chosen spatial point b, and then of a rotation with its axis through
Kb, where the given transformation changes b into Kb, First apply a
translation, axis bKb and parameter bKb; which is always a possible
transformation. This translation converts b into Kb. The second trans-
formation, which brings all the points into their final positions, must leave
Kb unchanged. Hence the transformation must be a rotation with its axis
through Kb.
(2) Applying the principles of § 2 75 below with regard to small
displacements, it will be easy to see that any small transformation is equi-
valent to the combination of a rotation and a translation with their axes
through any arbitrarily assigned point b.
464 KINEMATICS IN THREE DIMENSIONS. [CHAP. VI.
274. Commutative Law. The operations K and | performed succes-
sively on any point x are commutative, that is \Kx — K\x.
For let e, e,, ^, ^ be a unit normal system, e being the spatial origiiL
Let the transformation K have eei as axis, and S, a as parameters.
Then if x = ^e + fi^i + ^^ + f,^,
Kx = f f cosh - + fi sinh ~) ^ + f f sinh - + f i cosh - j e^
+ (fi cos a + fs sin a) ^2 + (ft cos a - ft sin a) ^.
Also [c£ § 240 (5)] \e = — ie^e^, |«i = — iee^, |^ = + icei€i, l^s = — ieexe^\
S S S S
and ir« = cosh - e + sinh - 6i, -E'ci = sinh - 6 + cosh - e,,
7 7 7 7
^^2 = COS a . ^2 — sin a . ^8, ^^ = sin a . e, + cos a . e^.
Hence Keei = KeKei = eei, and Ke^ = e^.
Thus |a?= -i(fCi + fte) ea^s - i (f a^a - ffl«s)««i ;
and if |ic = — i (f -iTci + ftiT^) eg^ — t (ft^^ — f»^^) «^-
Now ^Kei + ftif6 = (f cosh--l-ftsinh-Jei + ff sinh- + ft cosh -jc,
and ft^^i — ft^«8 = — (ft sin a + ft cos a) e, + (ft cos a — ft sin a) e^.
Hence by substitution and comparison
\Kx=:K\x.
276. Small Displacements. (1) Two finite congruent transforma-
tions, when successively applied, produce in general different results according
to the order of operation. If however each transformation be small, and
squares of small quantities be neglected, the order of operation is indifferent.
This is a general theorem, which holds for any linear transformation what-
ever. Thus let 616^^4 be any four reference points, and let x be transformed
into Kx by the transformation
ft' = (1 + an) ft + ttiafa + Aiaft + a^ft,
ft' = a«ft + (1 + a«) ft + Oaft + Ojift,
with two similar equations ; where all letters of the form otpp or a^ are small,
and their squares and products are neglected. Again, let Kx be transformed
into K'Kx by the transformation
?>" = (1 + /3u) f/ + /3^; + /3^,' + fir^:,
with three similar equations for ft", ft", ft"; where all the letters of the
form /Spp or /Sp, are small and their squares and products are neglected.
Substituting for ft', ft', etc. in the equations for ft", ft", etc. we find
ft" = (l + «u + Ai)ft + (aia + /9«)ft + (ai8 + A.)ft + (ai4 + A4)ft,
with three similar equations.
274—276] SMALL DISPLACEMENTS. 465
It is obvious firom the fonn of these equations for the co-ordinates of
K'Kso, that K'Kx= KK'x.
(2) Let a point ob with reference to the normal system e, euC^.e^ (spatial
origin e) be written in the form tie + 171^ + ri^ + ri^.
Apply a congruent transformation, axis eei, parameters S and a. Let x
become Kx. Then, as in § 269 (4),
Kx = (17 cosh - + i7i sinh - ) e + ( wi cosh - + 17 sinh - ) ft
V 7 7/ \ 7 7/
+ (i7j cos a + 17, sin a) e^ + (17s cos a — 17, sin a) e,.
Assuming that S and a are small, this equation becomes
h \ ( h
(3) Accordingly if in any order small translations S^, S,, S, and small
rotations ai, a,, Os be applied with axes ee^, ee^, ee^ respectively, and as the
total result x becomes Kx, then
- »? + aiVi - Os^i ) «i + ( 17s -I- -
iir«= U + - 171] « + fi7i + - 17] ^1 + (i;2 + ai7s) «» + (%- «^8)^s.
\777/V7 /
+ [V9+-V + «ii;8 - a^Vij «i + fi/sH- - 17 + aii7i - a,i7jj 63.
276. Small Translations and Rotations. (1) The result (K) of
the three small translations of the preceding article by themselves is a
translation, having as axis the line joining e and -^i+^es + ^e^, and as
7 7 7
parameters ^/{Si^ -I- S,* + Sj'} and 0.
For let d=^^,+-«2+ -e,.
7 7 7
Then /r« = e+d, and /Td = ^^J=^^±^% + A
Hence every point on 6c2 is transferred to another point on ed ; and any
plane of the form edx is semi- latent. Therefore the resulting displacement
is a translation with ed as axis. Let S be the parameter. Then
- = sinh - = V{- (eKe \eKe)] = V— (^ \^d)
.V-(i|d).y/^l±|^*.
(2) The result (K) of the three rotations by themselves is a rotation,
having as axis the line joining e and a^ei + a^ + a^^, and as parameters 0 and
For let a = a,ei -I- a^ + OjCs.
w. 30
466 KINEICATICS IN THREE DIMENSIONS. [OHAP. TI.
Then Ke=:e, and Ka = a.
Hence every point on ea is unaltered by the resulting transformatioiL
Therefore the transformation is a rotation with ea as axis.
To find its parameter, calculate the angle a between the planes teeiU and
iKieeiO) ; this is the required parameter.
Let A=ieeia. Then KA^ — iKei.Kea^ — i (61 — 0^ + a^s)^-
Also (il |il) = -(a.» + a,«)=(iril liTil).
Hence AKA = {c, («i — Os^j + a^s)ea] ea = i («,' + ««*) ea=-i(A \A)ea.
Thus a = 8iii«=y (^Wp^^^
(3) By properly choosing Sj, S,, S, the line ed, which is the axis of the
translation, can be made to be any line through e, while at the same time the
parameter V{Si" + 82* + S3"} can be made to assume any small value. Similarly
the axis of rotation ea can be made to be any line through e by properly
choosing ai, a,, Oj, while at the same time the parameter v/{«i*+ ««* + «»*} can
be made to assume any small value. Hence it follows from § 273 (2) that the
combination of the three translations along ee^, ee^, ee^ and the three rotations
round the same axes may be made equivalent to any small congruent trans-
formation whatever.
277. Associated Ststem of Forces. (1) Let 8 denote the system
of forces
(777 J
Then it is immediately evident by performing the operations indicated
[cf. §§ 99 (7) and 240 (5)] that*
Kx =s a? + i \xS.
(2) Similarly if P be any plane, then KP=^P + %\.P\S.
For let P = \p. Then Kp^p + i \pS.
By taking the supplement of both sides of this equation
\Kp^K\p^KP^\p + %\\.pS^P + i\(\p\S):=^P + %\.P\8.
Thus the system \S bears the same relation to the transformation of
planes that the system 8 bears to the transformation of points.
(3) It follows that, corresponding to every theorem referring to systems
of forces, there exists a theorem referring to small congruent transformations.
The system 8 will be called the associated system of the transformation.
Also since 8 completely defines the transformation, it will be adopted as
its name. Thus we shall speak of the transformation 8.
* This formula has not been given before, as far aa I am aware.
277, 278] PROPERTIES DEDUCED PROM THE ASSOCIATED SYSTEM. 467
278. Properties deduced from the Associated System. (1) If the
associated system be (with the previous notation)— ^1^3 + — 6^1 + —6i€^, the
ry 7 7
transformation is a translation along the line tf (S^^i + Si6^ + Sg^) of para-
meter V{Si' + Sj' 4- S,*}. Hence if the associated system be the single force F,
where F is anti-spatial, the transformation is a translation of axis \F and of
parameter V{-^l-^}-
If the associated system be the single force {aiee^ + a^e^ + Og^s), the trans-
formation is a rotation round this line of parameter Vfoci^ + Os' + 03'}. Hence
if the associated system be the single force F, where F is spatial, the trans-
formation is a rotation round F of parameter V{— -^1-^}-
(2) Hence the condition that the transformation ^S be a translation or a
rotation is {SS) = 0. The additional condition, that it be a translation, is that
(iSf|iSf) be positive, and the additional condition, that it be a rotation, is that
{S\S) be negative.
(3) The system S can be reduced [cf. §§160 and 162] to two forces
F + F', in such a way that either one force is in a given line, or one force
passes through a given point and the other force lies in a given plane. If
both F and F' are spatial, the transformation has been reduced to two
rotations round the two lines. If one (or both) of the forces be anti-spatial,
instead of a rotation round the line of that force a translation along its polar
line must be substituted
Since the given point may be assumed to be spatial, and the pole of the
given plane may be assumed to be a given spatial point, it follows that it is
always possible to reduce a small congruent transformation to a rotation
round an axis through one given spatial point and a translation along an axis
through another given spatial point. The two given points may be chosen
to coincide.
(4) The axis of the transformation is the axis of the system S.
Let ^ be a force on this axis, then
S^eF^-iit>\F.
If F^ \eei + fiee^ + vee^ + 'sre^i + pe^ + (reie^; then the condition,
X-cr + fjLp + pa = 0,
must be fulfilled.
Also S = (^+^)eei+(^/iA + ^p)6ej + (^i' + ^cr)e^
-h (0v - ^X) e^-hiOp- ^p) e^i + (Oa - ^v) e^.
Hence with the previous notation,
^-|-^«r = — «!, 0fi + <l>p =^ ^ CLi, ^1* + ^= — tts.
<?t»-^X=|. <?/>-^/.t = |. ea-<f>v^^.
80—2
468 KINEMATICS IN THREE DIMENSIONS. [CHAP. VI.
— UOLi — <j> — ff — — <f>CLi
Therefore \ = j, T^ , «r = 3[ , with four similar equations ;
where the ratio of 0 to ^ is given by
and a and S are put for V{«i' +08'* + as'} and V{Si' + Sa* + Ss»}.
The equation of condition gives two ratios for 0 : ^, and hence two lines
are indicated as the axis. One of these is F and the other is |^; the spatial
line is the axis.
The equation of condition for ^ : ^ can be given in another form. For
(S|S) = -a» + |, and (ag) = -2i H--^"^^^'-^"^|,
Therefore 20<l> (S |/S) + (<^« - 0*) % {88) = 0.
(5) The character of a small congruent transformation will be conceived
as completely determined by its axis and the ratio of 0 to ^ ; the remaining
constant simply determines its intensity. In other words, the two transfor-
mations, of which the associated systems [cf. § 160 (1)] are S and 'K8, are
of the same character with their intensities in the ratio 1 : X.
The performance of a given transformation \ times increases the intensity
in the ratio X : 1. For let 8 be the associated system.
Then Kx^x + t\x8, K*x^Kx'\-i\Kx8 = Ka: + %\x8^x-hii\ie8,
neglecting squares of small quantities.
Hence K^x = rr + Xi \x8.
279. Work. (1) Let any two spatial points a and 6 on a line ab be
transformed into Ka and Kb, let the angle between aKa and ob be '^ and
that between bKb and ob be ;^ then
. , aKa , . - bKb
smn cosysssmh cos^
or aKa cos -^ = bKb cos x* since the transformation is small.
To prove this proposition, notice that
• h ^^ — / — {(^Ka I aKa) _ V{- (aKa \aKa)]
^^^^ y "y {a\a)(Ka\Kaj ~(a\^ '
TT . 1^ <iKa a — (aKa \ab)
Hence smh cos 0 = . , .\. — ' ; ,., .
7 (a\a)'s/{—(ab\ab)}
279] WORK. 469
Therefore we have to prove that
^aKa \ab) ^ --(bKb | a6)
\a\a) " (6 16) •
Let the associated system of the transformation be S, where S = \(ab + cd).
Then
Ka = a + 1 \a8 = a + Xi |acd,
and aKa = iKa \ acd.
Hence (aKa \ ah) = (ab \ aKa) = i\ (ab . acd | a)
= iX [ab {{a\a) cd 4- (d \a) ac -I- (c | a) da\]
= i\ (a \a) (abed) = i (a \a) (aiS).
Therefore <?^^> = i (abS) = <-*^M) .
(a\a) {b\b)
We notice that if £> be the associated system of the transformation
^%(ab8)
aKa cos ilr = bKb cos v = ,, /- i-r-^ vt •
(2) Definition of Work. Let any point a on the line of a force of
intensity p be transformed by a small congruent transformation to Ka, so
that aKa makes an angle -^ with the force, then p . aKa . cos '^ is said to
be the work done by the force during the transformation.
It follows from the previous proposition that the work done by the force
is the same for all points on its line.
(3) Let b be another point on the force {F) so that F = ab. Then
p = ^{-{ab\ab)}.
Hence by (1) the work done by F during the transformation 8 is
-iiFS).
(4) Let the work done by any system of forces 8' be defined to be
the sum of the works done by the separate forces of the system. Thus let
8' = F'\'F' + F'' + etc.
Then the work done by 8^ during the transformation 8 is
- % {F8) - % {F'8) - 1 {F'8) - etc.,
but this is ''i(88y
We notice (a) that the work done by a system of forces during a small
congruent transformation is the same however the system be resolved into
component forces; and (fi) that the work done by the system 8' during
the transformation \8 is equal to the work done by the system 8 during
the transformation \8'; where \ is small, but the intensities of 8 and 8' are
not necessarily small.
If two systems be reciprocal, that is if (88') » 0, then no work is done
by either one during the transformation symbolized by the other.
470 KINEMATICS IN THREE DIMENSIONS. [CHAP. YI
280. Characteristic Lines. (1) Let the line joining any point
with its transformed position, after a small congruent transformation, be
called a characteristic line of the point: and let the line of intersection
of a plane with its transformed position be called a characteristic line of
the plane.
Thus if ^ be changed to Kx^ the line xKw is the characteristic line of
the point x ; and if the plane P be changed to KP^ the line PKP is the
characteristic line of the plane P.
(2) Let L be any line, and KL its transformed position, and let L intersect
KL, then L is the characteristic line of some point and also of some plane.
For let L and KL intersect in Ka, and consider the points a and Ka.
Since Ka lies on KL, then a lies on £, hence L = aKa. Thus L is the
characteristic line of the point a. Also consider the plane P = K^^a . a . Ka,
then KP = aKaK*a. Hence aKa (i.e. L) is the characteristic line of the
plane P.
(3) If S be the associated system, the characteristic line of any point x
is X \xS, and the characteristic line of the plane P is P { . P |£>.
(4) The locus of points x on the characteristic lines, which pass through
a given point a, is given by {axKax) = 0. This is a quadric cone through
the point a.
The equation can also be written {aKa . xKx) = 0.
Hence the characteristic lines of the points x are those characteristic
lines, which intersect the characteristic line aKa,
The equation can also be written (a \aS.x\xS) — Q\ that is [cf. § 167 (3)],
{axS){ax \S) - i(S8) (ax \ax) = 0.
(6) Similarly if AP be a characteristic line lying in the plane A, then
the planes P envelope the conic (APKAP) = 0, which lies in the plane A ;
that is to say, the characteristic lines in the plane A envelope a conic.
The equation can also be written {AKA . PKP) = 0. Hence the planes
P are such that their characteristic lines intersect the characteristic
line AKA.
The plane-equation of the conic can also be written
(APS) (APIS)- ^(88) (AP \AP) = 0.
281. Elliptic Space. (1) The Kinematics of Elliptic Space can-be.
developed in almost identically the same manner as that of Hyperbolic Space
[cf. § 268], only with a greater simplicity.
The absolute quadric being now imaginary, the four comers of the self-
corresponding tetrahedron in any congruent transformation must also be
imaginary.
I
280, 281] ELLIPTIC SPACE. 471
Let Oi, a,, a,, a4 be these four comers. Let the lines Oicc, and 0,04 be
real conjugate polar lines. Then a, and a, are conjugate imaginaries, and
so are a, and a^.
Thus let ei and e^ be two real quadrantal points on a^a^, and let ^ and &«
be two real quadrantal points on a^^. Then
ai = e *6i — 6*e,, aa = 6*«i — e *6a, 03 = 6 *ei — 6*^4, a4 = e*c, — 6 * ^4.
The transformation changes Oi into e~**ai, a, into e^a^^ (h i^^to ^^ct,,
a4 into 6'^a4. Hence any point « = 17161 + i/ae, + ri^z + 17464 is changed into
Kx == (% cos a + i/a sin a) 61 -I- (17a cos a — 171 sin a) 6s
+ (17, cos )9 + 174 sin)9) 6^ + (174 cos )9 — 173 sin/S) 64.
Thus (iTa; | Kx) = V + %* + '/s' + ^4* = (a? ! a?).
And (a? ! Kx) = (i7i» + 178') cos a + (V 4- 174') cos /3.
(2) Thus any point 17161 + 17,6, on 6163 is transferred through a distance 7a,
any point on 6^4 through a distance y/S. Similarly any plane through 616,
is transferred through an angle /3, any plane through 6,64 is transferred
through an angle a.
(3) The distance of Kx from 6462 is S, where
(BiB^Kx \eieJKx)
. S I (616
7 V \Kx
\Kx){exe^ 1 6,6a) '
But
{Sieji^x \eie^Kx) = (17, cos )8 + 174 sin /9)' -I- (174 cos ^ - 17, sin )9)» = 17,^ + 174*.
Hence sin - = * / / 1 v .
7 V (^k)
Thus the distance of Kx from eie^ is the same as that of x from 6162.
Similarly for the distances from ^4.
(4) Let a and /8 be called the parameters of the transformation. The
transformation will be described as the transformation, axis 6|6a, parameters
a, fi, or as the transformation, axis 6^4, parameters fi, a.
The transformation, axis 616a, parameters a, 0, will be called the transla-
tion, axis 6i68> parameter a ; or else, the rotation, axis 6s64> parameter a.
(5) Any congruent transformation may be conceived as the combination
of a rotation round and a translation along the same axis; or as the com-
bination of two rotations round two reciprocally polar lines; or as the
combination of two translations with two reciprocally polar lines as axes.
The distinction between translations and rotations, which exists in
H3^erbolic Spckce, does not exist in Elliptic Space.
472 KINEMATICS IN THREE DIMENSIONS. [CHAP. VL
282. Surfaces OF Equal Displacement. (1) The locus of points, for
which the distance of displacement in the transformation axis eie^, parameters
a, /9, is 7<r, is the quadric sur&ce
(w\Kai)
Hence employing the notation of the previous articles,
(vi" + Vi) cos a + (i7,« + V*) cos 13 = cos a (V + i/a* + V + V^^)-
Therefore (tji^ + rj^) (cos a - cos a) + (t;,' + f}^) (cos fi - cos c) = 0.
(2) The system of quadric surfaces found by varying tr is also easily
shown to be the system which remains coincident with itself during the
transformation. Its sections by planes through e^e^ or e^4, {i,e. by planes
perpendicular to e^^ or e^e^^ are circles.
283. Vector Transformations. (1) An interesting special case
discovered by Clifford* arises in Elliptic Geometry which does not occur in
Hyperbolic Geometry.
Let a congruent transformation, of which the parameters are numerically
equal, be called a vector transformation. Thus, with the above notation.
Hence [cf § 282 (I)] any point x is transferred through a distance 70.
Accordingly in a vector transformation all points are transferred through
the same distance, and similarly all planes are rotated round the same angle.
(2) Again, the line xKx is parallel to the axis 61^3.
For, taking a = ^,
xKx = - (i7i» + 17s') sin a . ^ea - (17,' + 174') sin a . 6^4
+ (171^4 - VzV^ SIR a . ^^ - (i7i'?8 + v%nd SIR « • «i«4
+ {v^Va + 'HiVz) sin a . ea^, + (171^4 - Vi'Hi) siR « • e-A
= - (^i' + 172^ SIR a . Ci^a - (17,' + 174') sin a . {e^e^
+ {ViVa - 178%) sin a {e^e^ - l^jC,) -I- (173174 + i7i';») sin a (e^ - \eA).
Hence xKx + \xKx = - (rfi^ + 17,* + 173' + 174') sin a (cie, + le^e^).
Similarly if a = — )8,
xKx — \xKx = — (x\x) sin a {e^e^ — l^jea).
Thus [cf. §§ 234 (2)] if a, a be the parameters of the vector transformation,
all points are transferred along right-parallels to eie^ \ and, if a, — a be the
parameters, all points are transferred along lefb-pcurallels.
Let these transformations be called right-vector transformations and left*
vector transformations respectively.
It follows therefore that any one of the lines parallel to the axis of a
vector transformation may with equal right be itself conceived as the axis.
* Cf. Clifford, Collected Papers, Preliminary Sketch of BtqwitemUms^ loe. dt. p. 870.
282 — 285] ASSOCIATED VECTOR SYSTEMS OF FORCES. 473
284 Associated Vector Systems of Forces. Let 12 be the unit
right- vector system of forces aieg+l^i^s [cf. §236 (1)], then the right- vector
transformation^ axis eie^ and parameter a, can be represented by
Kx = X cos a — sin a \xIL
This representation, unlike the preceding formula of § 277 and the subse-
quent formula of § 286, is not confined to the case when a is small. Let
R be called the associated unit system of the right-vector transformation,
axis ei^a.
Similarly if Z be the unit left- vector sjrstem ^^e, — \eie^, then the left- vector
transformation^ axis ^^ and parameter /3, can be represented by
K'w = w cos /8 + sin ^ \xL.
Let L be called the associated unit system of the left-vector transformation,
axis 6i6^.
286. Successive Vector Transformations. (1) If a right-vector
transformation and a left-vector transformation be successively applied, the
result is independent of the order of application*.
For let K and K' be the matrices denoting the vector transformations of
the last article only with different axes, namely BiB^ and &i V-
Then KK'x = K'x cos a - sin a | (K'x . R)
= a;cosacos^ + cosasin^|a7Z — sinaco8/8| xR
— 8inasin/3|(|d;£. R).
Also K'Kx = iTa; cos/8 + sin fi\(Kx . L)
= a: cos a cos^— sin a cos /9 \xR + cos a sin fi \xL — sin a sin iS [( jrciJ . L).
But \(\xL.R) = \\xL.\R== — xL.R] and similarly \(\xR , L) = xR . L.
But/by § 167 equation (26), since (iJZ) = 0 [cf. § 235 (4)], it follows that
(xL.R) + ixR.L) = 0.
Hence finally, KK 'x = K'Kx.
(2) Again, let jR and R' be both of the same name, say both unit right-
vector systems of the forms eie^ + leiBa, e/ej' + I^V; and ^©t K and K^ be the
corresponding matrices.
Then
K'Kx = a: cos a cos /8 - cosa sin /8 \xR' - cos /8 sin a \xR + sin a sin /8 \(\xR . R).
Now \(\xR.R)^^xR.\R^-xR.R = -(RR)x + xR.R\
Hence K'Kx = x cos a cos fi — cosa&mfi \xR' — cos yS sin a ^xR
— {RR) sin a sin /3 . a; + sin a sin iS . xR . jR'.
* CI. Sir R. S. Ball, loe. eit. p. 870.
f
} 474 KINEMATICS IN THREE DIMENSIONS. [OHAP. YI.
Similarly
KK'x = X cos a cos fi — cos a sin ^ | xR — sin a cos fi \ xR — sin a sin /3 . xR . if.
Hence K'Kx is not equal to KK'x,
Thus two vector transformations of the same name (left or right) applied
successively produce different results according to their order*.
(3) The resultant transformation, which is equivalent to two successive
vector transformations of the same name, is itself a vector transformation! of
that name.
Let R and R! be the two unit associated right-vector systems with any
axes, and let a and )9 be the respective parameters. Then it is proved in
the preceding subsection that
KK 'a; = a; cos a cos /3 — cos a sin ^ \xRl — sin a cos ^ \ xR — sin a sin /3 . xR . £'.
Let K^x be written for KK'x.
Also let ^ be a point at unit intensity, so that (^ |^) = 1.
Then {x \Kix) = {x \x) cos a cos ^ — sin a sin fi [x \{xR . i2')}.
Now x\{xR,R)^x.{\xR,\R')^x.{R\x,R) = x.R\x,R\
Again iJ = 6iea+ 1 61^3 = ay + \xp^ where p is the unit point on the right-
parallel to 6162 through x and normal to a;, so that {x \p) = 0.
Similarly R = ei'e^ -\-\eiei' ==xp' -\-\oop\ where p is a similar point such
that xp' is a right-parallel to ^iV-
Then iJ |a? = a5p |a7 = (x\x)p -- (x\p) x = p.
Hence x \(xR ,R')=^xp (xp' + \xp') = (xp \xp') = ^ (RRy
Thus (x \Kix) = cos a cos /8 — i (RK) sin a sin fi.
Accordingly, if ytr be the distance through which x is transferred,
remembering that (Kix\Kix) « (a? |a?) = 1, we find
cos a = cos a cos )9 — J (RRf) sin a sin fi.
Therefore the resultant distance of displacement of x to KiX is the same
for all positions of a;. Therefore [cf § 283 (1)] the transformation is a vector
transformation of which the parameter is a.
m
The proof is exactly similar if the two component transformations are
left-vector transformations.
(4) It now remains to be proved that the resultant transformation is
of the same name (left or right) as the component transformations. The
method of proof will in fact prove the first part of the proposition also.
It is easy to prove that, with the notation of the previous sub-section,
KK'x = cos (7 . a? — cos a sin ^ .p' — sin a cos /8 .p + sin asin )8 \(xp'p)^
and K'Kx » cos a . a? — cos a sin /9 . p' — sin a cos iS . p + sin a sin )8 1 {xpp^y,
* Sir R. S. BaU, loc. eit f Of. Sir B. S. Ball, loe. cU,
285]
SUCCBSSIVS VECTOR TRANSFORMATIONS.
476
Then, if JT, stand for KK\ we find
xKiX s ~ sin a cos /9 . ^ — sin yS cos a . o^' + sin a sin /9 . \p'py
since x \{xp'p) = {x \p) \xp' -{-{x \p') \px '\'{x\x) \pp = \p'p.
Now let y be any other point, and let q and ^ stand in the same relation
to 2^ as do j> and p' respectively to x. Then
yK^ » — sin a cos /8 . yj — sin ^ cos a . yg^ + sin a sin /8 . | g^g.
But «p + |ap = yj + |y?, «p' + l«!p' = y?' + |y?',
and by § 286 (2) pp' + \pp' = qq + Igrj'.
Hence xKyX + |a?-Kia? = yJf ly + [y-'fiy.
Therefore yiTiy and xKiX are right-parallels, which was to be proved.
(5) These theorems, due to Sir Robert Ball, have been proved analytically
in order to illustrate the algebraic transformations. They can however be
more easily proved geometrically.
For consider two vector transformations of opposite names applied
successively. Let the right-vector trans-
formation transfer a to 6, and the left-
vector transformation transfer 6 to c [cf.
fig. 1]. Complete the parallelogram ahcd
[cf. § 237 (3)]. Then the left-vector
transformation transfers a to (2, and the
right -vector transformation transfers d
to c.
Hence in whatever order they are applied the same ultimate result is
reached
But [cf. fig. 2] if any other point a' is transferred by the same
Fio. 1.
Fio. 2.
476 KINEMATICS IN THREE DIMENSIONS. [CHAP. YI.
combination of transformations successively to V and to c\ ad is not parallel
to oc, since a& and aHI are not parallels of the same name as ho and 6V
[cf. 237 (4)]. Hence the combination is not itself a vector transformation.
Consider again two vector transformations of the same name, say both
right-vector transformations, call them Tx and T,.
Let Tx transfer a to ft, a' to V ; let Tj transfer 6 to c and h to d [cf. fig. 2],
Then ah^a'U and 6c=6V, and the pairs of lines are parallels of the same
name. Hence by § 237 (4), aV and ac are equal and parallels of that name.
Hence the resultant of the combination is itself a vector transformation
of the same name.
But a parallelogram cannot be formed in which the opposite sides are
parallels of the same name. Hence the combination of T^ first, T^ second,
gives a different result from that of T^ fiirst, Ti second.
286. Small Displacements. (1) The theory of small displacements
in Elliptic Space is the same as that in Hyperbolic Space [cf. §§ 275 6^ seqq\
Let S = okfiie^ + «4a^«i + a«ei64 + OLue^ + «! A^a + an^* .
where the coefficients aia, etc., are small.
Then Kx = a; + \xS, gives
Kx = (f - ttiaf 8 - a„f, - tti^f 4) ex + (f , -I- ax^x - a«f a - olJ^^ e^
+ (f» + Oa^x + olJ^2 + fhii) ^3 + (^4 + auf + aj^t + «,«fs) e^.
This is the most general type of small congruent transformation.
(2) Also K\x=' \Kx. Let P be any plane, then
\KP^K\P^\P-\{\P.8).
Hence ZP = - |/i'|P = P + ||(|P.Bf) = P- KPjfi).
Thus |fif bears the same formal relation to the transformation of a plane P,
as S does to that of a point p,
(3) It follows, as in Hyperbolic Space, that corresponding to every
theorem referring to a system of forces there exists a theorem referring to
small congruent transformations. The system 8 will be called the associated
system, and S will be used as the name of the transformation.
(4) Thus if S=^F+F\ the transformation S is equivalent to two rota-
tions round F and F\ of parameters ^(F\F) and ^(F'\F') respectively.
Every transformation possesses two axes, which are found as in Hyperbolic
Space.
(5) Also [cf. § 160 (2)] if 6 be any point, S = p6 + P|6. Thus the
transformation is equivalent to a rotation, parameter V(p^ \pf>X round the
axis JP&, and a rotation, parameter ^/(P \b \. P \b), round the axis P\b: this
last rotation can also be described as a translation, parameter V(P \b\.P |6)>
along the axis b \P.
286] SMALL DISPLACEMENTS. 477
(6) Let any two points a and 6 on a line ab be transferred to Ka and
Kb ; let the angle between aKa and ab he 0, and that between bKb and ab
be <l>, then
. aKa ^ . bKb .
sm . cos 0 = BID . cos 0,
7 7
or, since the transformation is small,
aKa . cos 0 = bKb . cos ^.
This theorem is proved (as in Hyperbolic Space) if we prove that
{aKa\ab) _{bKb\ab)
{a\a) - {b\b) •
Now let the associated system S be written in the form X (oft + cd).
Then Ka = a-\-\aS=:ii,-\-\\acd,
and aKa = Xa \acd = \(a |(2) |ac + X (a \c) \da + X (a |a) \cd.
Hence {aKa \ab) = {db \ aKa) = X (a | a) {abed).
{bKb \ab)
Therefore (?^l ^) ^ ,, (,^) ^
(a |o) ^
{b\b) •
(7) The definition of work is the same as in Hyperbolic Space.
The work done by a force is (by the last proposition) the same for all
points on its line.
If "KS be the associated system of a transformation, where X is small and
5 is not necessarily small, and S' be any system of forces, then the work done
by S' during the transformation \S is \{8Sy This is equal to the work
done by S during the transformation \3\ The proof of this theorem is
exactly as in the case of Hyperbolic Space.
If two systems S, 8' are reciprocal, so that (SS') = 0, then the work done
by 8 (or 8^) during the transformation X^' (or XS) is zero.
CHAPTER VII.
Curves and Surfaces*.
287. Curve Lines. (1) Let the Space be Elliptic (polar) and of three
dimensions, though the resulting formula will in general hold for Hyperbolic
Space. Let any point x be represented, as usual, by f ,ei + fA + f ^ + f A,
where the co-ordinate points ^, e,, ^, e^ form a unit normal system.
Now let the co-ordinates of x, namely fi, fst fs> ^4» he functions of some
variable t. Then, as t varies, x traces out a curve line.
When T becomes r + St, where St is indefinitely small, let x become
x + xSt.
Then obviously iP = |i«i+^A + fjCi + ^<e4. Let Xi stand for x-hxSr. Let
X, X etc. be derived in regular sequence by the same process as i is derived
from X.
(2) Now as X changes its position to o^i, it might change its intensity
as expressed in the above notation. Let it be assumed that the intensity
of X remains always at unit intensity.
Hence (a?|a:) = l, {x\x) = 0, (x\x)'\-(x\x)=^0, and so on, by successive
differentiations. Hence ^ is a point on the polar plane of x.
Also the same variation of r to r + Sr, which changes d? to o^, will change
a?i to aji, where aji = a?i + fl&iST = ir-l-2*Sr + a5(ST)», and will change aj^ to a^,
where «i = a? + SxSt + Sx (StY + x (Sr)*, and so on.
(3) Let &r denote the length of the arc^pj, then
-"^7V(f^Sgfe)-^'-*|->-^
= V{(^ I*) - (^ \iY] St = V(* \x) St.
Therefore ;t-=^7V(^1^)«
* ThiB applieatton of the CalcnlciB of Extension has not been made before, as far as I am
aware.
287, 288] CUBVE lines. 479
(4) Again, let Se denote the angle of contingence of the curve at x
corresponding to the Bmall arc S<r. Then Be denotes the angle between xxi
and a^Xf. Hence by considering the triangle xxjpe^, we find [cf. § 216 (2)]
5i^ o,-« 5i^ /(^ox^xx^x^
{XX^ \XX^ {XiX^ 1^1^) '
Now xx^x^ = XXX (Bry ;
and (^lOTs l^i^s) =^ (^^1 1^^) = (^ 1^) (BtY, ultimately.
_ » J(xxx \xxx) ^ Jlxibx \xxx) ^
Hence he = \ . . .>, or = \. ,' — - St.
(xx \xx) \x \x)
Also we may notice [cf. subsection (2)] that
{xAx \xxx) =1, 0, — {x \£)
'-{x\x\ {x\x\ {x\x)
= (i? \(b) (x \x) - {x \xf - (x \±Y
= {xx \&x) — {x \xf.
Hence Be = ^K^^ If^) ^ (^ l^^l g^.
(5) The tangent line at ^ is the line xx^, that is, the line xx.
The normal plane at a? is the plane through x perpendicular to xx^. This
plane is the plane x \xxi, that is, x \xx.
Now x\xx^{x \x) |a? — (a? \x) |A = — \x.
Hence \is is the normal plane of the curve at x.
The normal plane at Xi is therefore \x + xBr.
Monge's ' polar line ' of the curve at a? is the line of intersection of the
two planes, that is, the line \xst.
(6) The osculating plane of the curve at a? is xx^^, that is, the
plane xxx.
The neighbouring osculating plane at Xi is xxx + xxxBt.
The angle between these two planes, namely, B0, is the angle of torsion
corresponding to the arc &r. If the first plane be P and the second plane
be Q, then
FQ = i€XX . XXXOT = {XXXX) XXOT.
Hence
V (P I P) (Q I Q) (oixx \xxs) {xxx \xxx)
288. Curvature and Torsion. (1) Let the 'measure of curvature' or
the * curvature ' of the curve be defined to be the rate of increase of e per
unit length of a, and let it be denoted by - .
480 CURVES AND SURFACES. [CHAP. VII.
Let the 'torsion' of the curve be defined to be the rate of increase of 0
per unit length of <r, and let it be denoted by — .
Th 1 _ de _ 1 >sj{xxx \xxdi) _ 1 ^{{xx \xx) - {x \xf\
p^da y (x\x)^ ""7 (x \x)^
1 _ dtf _ 1 (xxxx)
K do- "" 7 {xdsx \xdMt) *
-T 11 (xxxx)
Hence ~r = i rr^ •
P^K 7* (X \xy
The condition, that a curve is plane, is - = 0 ; that is, {xxxx) = 0.
(2) Newton's geometrical formula for the curvature still holds. For
let &r be the distance between the points x and x + hx, and let Si; be the
perpendicular from ^ + So; on to the tangent xx.
The. [ef. 5 226 (I)] h.^<^0).
But hx^X + (thT + \x(hT)\
Hence
hv _ 1 1 HxxSt \xxx) __ 1
(3) The normal planes envelope the 'polar developable/ of which the
edge of regression is the locus of intersection of three neighbouring normal
planes.
The polar line \xx is a tangent to this edge of regression at the point
corresponding to x.
The three normal planes 9X x,Xi,(t^ are
\A^ |i + |a.8T, |i? + 2 |a5.8T+.|a:.(ST)'.
Hence the point on the edge of regression of the polar developable which
corresponds to a; is \xxx. Let the distance, pi, of this point from x be called
the radius of spherical curvature.
(* •.•••\
XXXX)
AUtJU CUB - = ,, , ; .
7 fsjyxxx \xxx)
(4) The polar line \xx meets the osculating plane in a point called the
* centre of curvature.'
This point is xxx \d!x.
The distance of this point from x is called the radius of curvature —
which is not in general equal to the inverse of the measure of curvature
except in parabolic geometry.
289] CUBVATURB AND TORSION. 481
In order to find the radius of curvature, which will be called /Og, we
notice that \±x is perpendicular to ocxx. For the line through the point
XXX perpendicular to xxx is the line \xxx. \xxx = \(xxx. xxx)^-- (xxxx) \±x.
Hence neglecting the numerical factor, this is the line \xx.
Therefore p, is really the distance of x from the polar line \xx.
But the distance of x from any line F is
. _ f / (xF\xF) \
Now X \xSi = (x \x) \x — (x \x) \x = (x \x) \x.
And {\xx . \\xx) « (xx \xx).
„ . p. /(x\xy(x\±) (x\x)^
Hence 8in^=./^ /'^\'Jr^= //"i.-n-
7 \ {xx\x<id) V(^^|ara?)
Therefore ^ = cosec» ^ - 1 = cot« ^ .
P 7« 7
Hence P = 7 tan — .
7
(5) The principal normal is the normal line in the osculating plane,
that is the line,
xsbx \x =s {x \x) XX -{- (x\di) XX + (± \x) XX = (d)\x)xx -h {it \x) xx.
The binormal is the normal perpendicular to the osculating plane, that is
the line x \xxx.
It will be usefril later to notice that the intensity of xxx \x is
V{(^ 1^) i^xx \xxx)}.
For if P and Q be two planes, then (PO|PO) = (P|P)(QI(2)-(P|(2)»; and
by writing xxx for P and \x for Q the result follows.
289. Planar Formulae. (1) The complete duality both of Elliptic
and of Hyperbolic Space allows formulse similar to the above to be deduced
from the plane equation of a curve.
Let P denote the plane Vyfi^^^ — v^e^^ + v^e^^ — v^eie^. Let »i , v,, t;,, v^
be functions of one variable r. Then the plane P, as it moves, envelopes a
developable surface, of which the curve under consideration is the edge of
regression.
Let Pi = P + i^ST, where P — Vie^e^^-hetc, and let P be derived from P
as i^ is from P, and so on.
Let P, = P + 2P.ST + P(iry, Ps = etc. Let (P |P) = 1, so that
(P\P) = 0,(P\P) + {P\P)^0.
(2) The point on the curve corresponding to P is the point PPiPg, that
is the point PPP. The tangent line in P is the line PP.
W. 31
482 CURVES AND SURFACES. [CHAP. VII.
The angle of torsion 80 is given by
(3) The angle of contingence Be is the angle between the lines PPi and
PiPa- Hence by considering the stereometrical triangle P, Pi, P^ and
deducing the formula from that for a point-triangle by the theory of duality,
we find
s.-«in;i.- / (PP^P^ \PPiP^) _^/(PPP\PPP) .
' " V (pp:\jm{PiP~,\p^p>) {p \pj '
_^[(PP\PP)-(P\Py},
(P\P)
(4) The length of the arc Ba- is obviously found frt)m the analogous
formula to that which gives the angle of torsion in the point-equation.
Thus ^s^d^^mmsr
TKpn 'k - 'J{PPP\PPP) _ ^[{PP\PP)-{P\Py] .
d0- (P\p)i - {P\P)i
da t (PPPP)
and
dti~(PPP\PPPy .
„ da y(PPPP){P\P^
Hence p = ^= (PpP\FpP^
K =
da y(PPPP)
de^(ppp\pppy
(5) The point |i^ is a point in the plane P, such that any line in P
through 1^ is perpendicular to the tangent line PP. For the point, where
|Pi^ meets P, is the point P|Pi^ = (P |i^)|P-(P|P)|i^ = -(P|P)|l^, and
this is the point {P, neglecting the numerical factor.
Thus \P is the point on the principal normal distant ^wy from the
point PPP.
The normal plane is the plane PPP \PP, this can easily be shown to be
the plane (i^ |i^) -P + (P 1^)2 P - (i^ |i^) A
290. Velocity and Acceleration. (1 ) Some of the main propositions
respecting the Kinematics of a point in Elliptic or in Hyperbolic Space can
now be easily deduced. It must be remembered, that in such spaces the idea
of direction, as abstracted from the idea of the frilly determined position of a
line, does not exist. In Euclidean Space all parallel lines are said to have
the same direction. Again, in considering the small displacements which
a moving point has received in a small time Br, we must remember that
290] VELOCITT AND ACCELERATION. 483
the propodtions of Euclidean Space are applicable to infinitely small figures
in Elliptic or Hyperbolic Space.
. As in the case of cm*ved lines the reasoning will explicitly be confined to
Elliptic Space.
(2) Let the variable r of the preceding sections be now considered to be
the time. Then the point x in the time hr has moved to the position
X + dihr. The line joining these points is the line of the velocity, and the
length of the arc traversed is its magnitude. Hence the linear element xx
represents the line of the velocity, and its intensity represents the magnitude
of the velocity. Hence xsb may be said completely to represent the velocity.
Hence at a time r -^Zr the linear element {x + xhr) {ab + oSSr), that is
xx-^xx .ir, represents the velocity. Therefore, remembering that the
propositions of Euclidean Space can be applied to the infinitely small figures
which are being considered, the linear element xct completely represents the
acceleration.
(3) It is also obvious, from the applicability of the ideas of Euclidean
Space to small figures, that two component velocities Vi and v^ along lines
at an angle o» are equivalent to one resultant velocity of magnitude
V(vi' + ^% + 2t;iV9 cos a>), making an angle 0 with the line of Vi ; where
Vi _ ^« _ ty
sin (w — d) ~ sin 6^ ~ sin » *
The same theorem holds for accelerations.
Thus if xab and xdf denote two component velocities of the point Xy then
xx-\'Xd: represents the resultant velocity [c£ § 265 (4)] ; and if xx and xW
represent two component accelerations, then xx+xx' represents the resultant
acceleration.
(4) The magnitude of the velocity is &, or y^{x\x). It will now be
shown that the acceleration is equivalent to two components, one along the
X \x I
wuA^uuu ui luoj^uiuuuv? ff, %ji 7 //» I «v> ^^^ ^^6 other along the principal
normal of magnitude — .
xdi
For a unit linear element along the tangent is . . , .. ; also [c£ § 288 (5)]
XXX \x
a unit linear element aloni? the principal normal is ± ,,, . . .v / .-i ..»m .
*^ * vK^ \^) (,^!ca5a? \osxx)j
, . , , , ... .{sb\x)xx-{x\±)xx
which can also be wntten ± \(! . . .v, ... t' ...v, .
V K^ 1^) (^«^* \xxx)\
^_ (df\x) a^ /{{xxx \xxx)\ (x \x) xx-jx \x) xx
Also yxx^y^^^^ . -^^^i^^ + ^y I ^^ 1^^ I ^j^^ 1^^ ^^„ |^„^j
..XX a* XXX \x
~ ^ ^(a]±) " J ^{(x\x){xAx\xxx)] '
31—2
484 CURVES AND SURFACES. [CHAP. VU.
Hence the acceleration is equivalent to two components as specified
above.
(5) Let the point x be called the velocity-point of a?, and of the accelera-
tion-point of X, The velocity of a? is directed along the line fix)m x towards i
and the acceleration towards x. Furthermore, if v be the magnitude of the
velocity and a of the acceleration, then,
v=7\/(^|^))
and a = 7 \/(^ |^) = 7 V{(^ |3?) — (a? \xf\ = 7 V{(^ 1^) — (^ l^^}'
Hence 7*a' + 1;* = 7* (^ |^).
29L The Circle. (1) To illustrate the formulae of §§ 287. 288 con-
sider the circle, radius a, in Hyperbolic Space [c£ §§ 216 (6) and 259].
Firstly let the centre be the spatial point e. Let 6, ^i, e,, ^ be a system
of normal points at unit intensities, spatial and anti-spatial respectively, and
let ee^e^ be the plane of the circle.
Then if a? be any point on the circle [cf. § 216 (6)], and ^ be the parameter
T of §287(1),
X = 6cosh - -I- «i sinh - cos <i + eg sinh - sin <6,
7 7 7
X = sinh- {— ^1 sin ^ + ^^cos ^},
X = sinh - {— Bx cos 6 — «« sin 6} = e cosh — a?.
Therefore -rj = 7 V— (^ I ^) = 7 sinh - :
a<p 7
and a- = 67 sinh - ; and the length of the whole circumference is 2iny sinh - .
7 7
Also
#V ^tf 4if
xdi'x = sinh - («i cos ^ + ^jsin ^) . sinh - (- «i sin 6 + e, cos A) . ecosh -
7 7 7
= ecjea . sinh* - cosh - .
7 7
Hence -= . = ^ ..' ^ = cosh - .
d^ — (a? |a;) 7
The normal plane is \x^i sinh - \ee^^ sin ^ + ee^e^ cos ^}.
All the normal planes pass through the line ee,, drawn through the centre
perpendicular to the plane of the circle.
The measure of curvature is given by
1 1 Jixxx I (cxx) 1 , , a
_ = - L-b^ — _ coth - .
P 7 {rx\xy 7 7
291] THK CIRCLE. 485
Hence p = 7tanh-.
7
1
When the radius is infinite, the measure of the curvature becomes - , and
7
this is its least possible value ; when a is zero, the measure becomes infinite.
The binormals obviously all intersect in ^, and form a cone with an anti-
spatial vertex.
(2) Secondly, let the circle have an anti-spatial centre, e„ and let it lie
in the plane eeie,; where e,ei,e^,e^ form an unit normal system of points
with e as spatial origin. Then [c£ § 259 (5)] the circle is a line of equal
distance from the line e^i.
Let P be the distance of a point x on the circle from this line. Then
smh - = / -/- , \ / - . \ = w(xeei \xeeX
when X, e, ei are at unit intensities, spatial and anti-spatial.
Hence if a; = fc + f i6i + f^ea,
we have f * - f i» - f,' = 1, and fa = sinh ^ .
Hence we may put
f = cosh - cosh </), f 1 = cosh - sinh <^.
Q Q Q
Thus x^e cosh - cosh <6 + «i cosh - sinh A + «> sinh - ,
7 7 7
o
(b =s cosh - [e sinh 4> + ^ cosh <^},
Q
X = cosh - [e cosh ^ + ^i sinh ^}
= a? — ea sinh - .
7
The normal plane is given by
Idp = t cosh - 1— e^e^ sinh 6 — ee^ cosh 6],
7
The principal normal to the curve is the intersection of this plane with
eeiSt] and it is perpendicular to the axis, since \x contains the polar line of the
axis, viz. e^.
This normal meets the axis ee^ in the point eei |x, that is in the point
Ci sinh <^ + c cosh <^. Call this the point y, then y = « sinh <^ + ei cosh <^.
Let he' denote the distance between y and y + y8<^.
486 CURVES AND SUBFACES. [CHAP. VII.
Then ^ = 7V-(*|i>) = 700sh^,
Hence [cf.§ 262 (3)1 ^= cosh ^, «r = «r'coeh^.
" d<r 7 7
Aeain aidsx = — 66,6, cosh' - sinh - .
« 7 7
, cosh' - sinh - „
Hence , i = 'a = sinh - ,
^* cosh^ ^ 'y
7
and - = - tanh - .
P y 7
When fi is infinite, p = 7 ; and when /9 is zero, the curvature is zero. In
this latter case the curve is identical with the straight line which is the
axis.
(3) Thirdly, let the centre of the circle be on the absolute, so that the
curve is a limit-line. Let e be a spatial point on the curve, and let eCi be the
normal at e, and eCie^ be the plane of the curve ; also let 0, ^i, es, ^ be a
normal system of unit reference elements. Then from § 261 (3), the limit-
line becomes the section of the limit-surface
(ai\x)-{iD\(e+ei)]*
by the plane e^ie,.
Thus, if w be of unit intensity, it can be exhibited in the form
^ = (1 + ^^)6 + i^ei + ^^.
Then, if ^ be the parameter t of § 287 (1),
Hence ;7Z = 7 V" (^ 1^) = 7» <^'^0y.
d0
Thus we can write
1 cr» . <r
where <r is the length of the arc from e to a?. Then writing x' aad of' for i
and X with this new variable a^
fl^=-i« + ^ei + -^, a^' =« -5 (« + ei), and asa^a^' = - — ^«^.
Hence -— = - = -.
aa p ff
292] THE CIRCLE. 487
This agrees with the deduction above, that a circle of infinite radius is of
curvature -.
7
Again let y be a point on a second limit-line with the same centre as
the first one, and at a constant normal distance B from it ; so that, if x and y
be corresponding points on the first and second limit-lines respectively, the
line ay is normal to both curves, and xy is equal to S. Then if y be of unit
intensity, it is easily proved that
y = exp ( — j X -^ sinh - (e + 6i).
Hence if cr be an arc of the x curve and <r' an arc of the y curve, and if
a- be the independent variable,
y=exp(--ja^.
Hence ^ = 7 V-(y |y) = 7exp(---) V-(a?'|a?0 = exp(--j.
Hence <r' = exp( — j.cr.
7>
7'
This result ia proved by both Lobatschewsky and J. Bolyai.
292. Motion of a Rigid Body. (1) Let S stand for the system
and let &St be the associated system of the transformation which would dis-
place the body from its position at time r to its position at time r + St. Let
any point of the rigid body be x at time r and x + &Bt at time r + St.
Then x + xSr = a? + \xS . St.
Hence fl& = + |a?/S.
Let 8 be called the associated cfystem of the motion of the body.
(2) The theorem of § 286 (6) can be stated in the form : the resolved
parts of the velocities of two points of a rigid body along the line joining
these points are equal.
Thus if 07, y be the two points, and xx and yy make angles 0 and <f> with
xyy then
^ cos 0 = ^ cos ^.
(3) The velocity of any point x of the moving body is
Hence the velocity of each point is perpendicular to its null plane with
respect to S,
488 CURVES AND SURFACES. [CHAP. VII.
(4) Again if & change to S + S . St at the time t + St,
then ^ = |a?S+|i?A = |a;S + li;liS=|a;S + a;S|S.
(5) It is obvious that all the theorems which have been enunciated with
respect to small congruent transformations hold with verbal alterations for
the continuous motion of a rigid body.
Thus if £1 be a vector system of the form OiOj + loiOj, then |/§ = ± S. Also
suppose that S is constant with respect to the time. Then
Thus XX is always zero and no point of the moving body has any acceleration.
Thus each point of the body is moving uniformly in a straight line. This is
a vector motion of the body.
293. Gauss* Curvilinear Co-ordinates. (1) Let x be any unit
point on a surface in Elliptic Space. Then the co-ordinates of x, referred
to any four reference elements, may be conceived as definite functions of
two independent variables, 0 and <f>. And the two equations, d = constant,
^ = constant, represent two families of curves traced on the surface.
(2) Suppose that the unit point x + Bx corresponds to the values O-^-Sff
and ^ + S0 of the variables. Then we may write Bx in the form
Bx = XiB0 + x^B<l> + i [x^^ {BOy + 2xiJB0B<t> + x^ {Bif>y] + etc.
Since the point remains a unit point, we see by making B0 and B^
infinitely small, and by remembering that the ratio of B0 to S^ is arbitrary,
{x\xi)^0^{x\a^).
Hence x^ and x^ are in the polar plane of x.
In order to exhibit the meanings of Xi, x^, x^, etc., let e^fi^^^ be a set of
four unit quadrantal points ; and let x = Sfe. Then oh — ^Si^^ a^ = 2oT«i
ou dip
«ii = 2 g^ e, and so on ; where the condition, f i« + f a« + f 3' + f 4' = 1 is fulfilled.
It will be an obviously convenient notation to write 5^ for a?i, and so
on, when occasion requires it.
(3) By diflferentiating the equations, (a?|a!i) = 0 = (a?|«j), with respect
to 0 and ^ successively, we obtain
(a?i \Xi) = - (a? \xu), (aJa iiCa) = - (a? \x^\ (Xj \x^) = - (a? |a^).
(4) The distance So- between x and a? + Sa? is given by
-7^ = sin* — = (a&c \xSa;) = (axx^ \xx,) {BOy + 2 (asTj |asr,) 8dS»f> + (xx, |a»B,) (Sij>y
= («i K) (Bey + 2 (as, ja:,) BeSif) + {x^ !«,) (S^)«,
293, 294] gauss' curvilinear co-ordinates. 489
(5) The tangent line to the curve joining the points x and a; + &>? is xhx,
that is x{xiB0 + x^<l>). Hence xxi and xx^ are two tangent lines to the
surface at x, and therefore the plane xxix^ is the tangent plane at x.
The normal at a; is the line x \xx,x^.
But X \xxiXi = {x {x^) \xa!i + (^ |^) \x^ -\-{x\x) |^a^ =: l^i^,.
Hence the line \x1X2 is the normal at x,
294. Curvature of Surfaces. (1) Let Bv be the perpendicular fix)m
the point a; + &r on to the tangent plane at x.
Then [c£ § 224 (4)]
Bp {xXiX^x)
7 '^/[{xXiX^ I xx^x^]
_ 1 (ocxiX^ii) (Bdy + 2 (xxyv^i^) B0B<f> + {xxjx^^ iB<f>y
ju \ \XiX^ \X\X2\
Let - be the measure of the curvature of the normal section at x through
P
the tangent line xSx. Then
= 11^)"= \/{a?ia?» kar,) . {{x, \x^)(S0)^ + 2 (x, Ix^) B0B(f> + (x^ \^2){B<l>y}
^ 2 Bp '^ {xx^x^i)(B0y + 2(xxiX^^)BOB<l> + (xx^x^{B<l>y
(2) Now seek for the maximum and minimum values of p, when the
ratio of B0 to S0 is varied. Let pi and ^ be the maximum and minimum
values found, and let B0i/B(^ and S^s/S^ be the corresponding ratios of
B0IB(f>.
Then f>ily*/{^^\^'^] and /98/7\/{a^^|a^^} are the roots of the quadratic
for f :
{{xx^x^i) ?- (a?i \Xi)} {(xx^XtX^) f - (ic, \xt)} - {(i»i?|fl!Bais) ? - («i !«,)}' = 0.
Hence — = i^'^f^i) (00^^^ ~ {p^w^ .
/>!/>« 7"{(^l^i)(«i|^)-(^k2)*}' '
1 ^ 1 ^{xy\x^){xx^x^x^)^-{x^\x^{xx^x^-2{^^
f^ P' y[(x,\x,)(x,\x,)^(x,\x,y\^
(3) The expression for — can be put in terms of (xi |«i), (ic, [x.^), (x^ \x^\
piRa
and of their differential coefficients with respect to 0 and ^.
For (xxix^u) (xxjX^x^) = (xxios^u IxXiX^c^)
1 , (x\xi) , (x\xt) , («?!««)
(X^\X), (Xj\Xi), (Xi\Xt), (Xi\Xn)
(xt\x), (xt\xi), (asala?,), («i|«a)
(«iik), («iiki), (a?ii|^), (a?u|«»)
490
CURVES AND SURFACES.
[chap. VXL
Similarly
{poxiX^x>^y =
Hence, since {x |a^) = 0 ss (a; \x^,
{auCiX^i)(cMPiX^^ (^|«,), («?i|«i), (a?i|«a)
(«i|^)» (asalaJa), («9|«b)
(«i|«ii), (a:aK), 0 i
+ {(aq l^i) (^ |«9) - {a^ |«J»} {(a?u |aia) - (a: |«?ii) (x \xji\.
(xi\x,), (a?i|a?.), (a?i|iri,)
(xi\x^), (a^alfl?,), (aSi|a?u)
+ {(«! |aa) («, |as,) - (x^ IxtY] {(j?u |a:u) - (x !«?«)»}.
Now write ^ (a?i |aJi) in the form (xi |a^X* ^^^ aS ^^ 1^^ ^ ^''^^ ^*^"^ ^^ 1^^'
with a similar notation for the differential coefficients of similar quantities.
Then by successive differentiations of the equations (a;|d;i) = 0 and
(x\x^) = 0, we find
(a?iki)==i(^k)i, («2|a«)=i(a^|«2)i, («i|«u)=4(«ik)B, («a|aTi)«i(a:«kX;
(«9 |«u) = k kX - i («^ k)»» (a?i l«») = («i kX - i («si kX-
Also (a?u|«B)-(«fif|«a)==(aik)ia-ikk)u--i(a?ik)«-
Hence ^K^k)kk)-"(^ik)'}'
(a?i |ai), («! k). k k)« - i k kX
kkX kkX ikkX
ik kX» k k)i - i k kX» o
(Xi\xi), kl^iX ikkX
kkX kkX i(«ikX
i(^\^)%» ikl««X» 0
+ {k 1^) k k) - k k)*} Ik k)* - k k) k k)
+ k kX« - ik l«8)ii - i(«i kXi}.
is expressed in the required form.
Thus
fVa
(4) Hence follows the extension to Elliptic and Hyperbolic Space of
Gtauss' theorem with respect to the applicability to each other of two small
elements of surface. It is evidently a necessary condition, that — should
be the same for each element.
hfh
296. Lines of Curvature. (1) By the usual methods of the ele-
mentary Differential Calculus it is easily shown that the ratios S^i/S^ and
B0t/^9 which give the directions of the lines of curvature (defined as lines
295] LINBS OF CURVATURE. 491
of maximum or of minimum curvature) through m^ are the roots of the follow-
ing quadratic for hOjh^ :
+ {(«il««)(a»»i«>a?ui) ~(«i l^)(««CiaWB)} (S0)* = O (i).
This equation can be put into another form.
For [c£ § 293 (3)]
{Xi \x^ (aucjx^^) - (xi \xi) (xx^x^ = {x |a?u) (xx^x^ - (x \xu) {xXyX^
= [{{oiXjX^ Xii - (xxix^i) x^}\x]=^ {(««a«i . Xj^Xjsi) \x],
by § 103 (8), equation (4).
But the product {{xxios^ , XiiXj^)\x} is pure [c£ § 101], and therefore
associative.
Hence {(xXiXt . a^a^a) \x] = (xx^x^ \x. XnXa) = (xiX^ XuXj^,
since xa^x^\x^(x\x)a>ia!i + (xi\x)x^'\-(x^\x)xxiSiXjXi, by § 293 (2).
Similarly (a^ {x^) (xXiX^Cu) — (xi \xi) (xx^ic^c^) = (xiX^sdjiXu),
and (x^ I Xf) {xxiX^c^ — {xi \x^ (xxjX^Cn) == (a?iav>WE»)«
Accordingly equation (i) takes the form
(x^x^^Cja) BO^ •¥ (XiX^Cu/Cn) B0B<l> + (xiX^^c^) 8^=0 (ii).
It easily follows from equation (i) that
(x, \x^) B0,80^ + (a?i la^a) {Be^BiJH + B0JB<t>,] +(x^\x,) 8<^S^ = 0.
(2) Letx-^Bx and x + S^x he any two neighbouring points to a; on the
surfince, where
Bx = x^Be + xj^, B^x = fl?i8'5 + x^B'^.
Then the angle ^ between the two tangent lines xBx and o^a? is given by
(a£x IxB^x)
*^y^ ^[(a^ \xSx) (wS'x I xS'w)]
. (g, |aO SdS'0 + (a;, |a^) (8gy^ + B'dB<t>) + (a;. |a!|) 8^y<^
-'^^ g^^SV '
where So- and S^<r are the arcs between x and « + &;, <b and x+i^w.
Corollary. The lines of curvature cut each other at right angles.
(3) Since (a; | &c) = 0 « (« | ^x), where &b and ^x are infinitely small.
Hence sin ^ = ^{(^P^^g^jg^ •
Therefore S<r8o'eixii^'"f>^{BaS^x\8xSrx}
= 7» (Sd8'<^ - y^S^) V{(«i |«i) («ii 1*0} . sin • ;
where a> is the angle at x between the curves ^.b constant, ^ = constant.
492 CURVES AND SURFACES. [cHAP. VlL
(4) If the curves, 6 ^ constant^ ^ » constant, be lines of curvature at
all points, then the equation for the lines of curvature must reduce to
Hence from subsection (1), equation (i),
and these equations must hold for all values of 0 and ^.
(5) Let — be the measure of curvature of the normal section through
Pi
xxi, and — of that through xa^; where the 0 and if> curves are lines of
Pa
curvature.
The radius of curvature of any normal section [cf. § 294 (1)] is given by
p Pi Pi
The angle y^, which the tangent line xhx makes with the tangent line xxi,
is given by
cosy* =
sin-^s
jj 1 _ COS* -^ sin* -^
P Pl P2
This is Euler's Theorem.
(6) The condition for the 0 and ^ curves being lines of curvature may
be put into a simpler form than that in subsection (4).
For we have {x \x^ = 0 = (ar |a?,) = (a^ \x^ = (xxiX^x^),
Hence [cf. § 293 (3)] (x |a?i,) = 0.
Hence since (x \Xi)^0=^{x 1X2) = (x |a^), either the three equations are
not independent and x^ can be written in the form
or a? is of the form v l^iavzaa.
Taking the latter alternative, and substituting in the equation
{xxiX^^,^ = 0,
we find (^^i^is l^i^s^])) = 0.
But the condition (P|P) = 0 cannot be satisfied in Elliptic Space by a
real plane area. Hence it implies, if the plane area is known to be real,
P = 0.
Thus we are brought back to the first alternative, namely XiX^Dis^O.
296, 297] LINES OP curvature. 498
If the space be Hyperbolic, the condition, (P { P) » 0, implies that all
the points on the plane P be anti-spatial, except its point of contact with
the absolute. Hence if a; be a spatial point, (wP) cannot vanish. So again
the first alternative is the only one satisfying the conditions.
296. Meunier's Theorem. The measure of curvature of the curve,
0 = constant, is found from the formula of § 288 (1). Writing — for it,
iP is given by
The measure of curvature of the normal section through ofXi is given by
Hence ^ « (^x k)* (^i^»Wi)
P V K^i^ii I ^i^ii) (^^ I ^1^)1
The osculating plane of the curve is xx^Xu. Let % be the angle between
this plane and the normal section, which is the plane x^ IXiX^.
_, {Xi \XiXi, l^^i^u)
COS X - ^^(^xx^x^i ka^i^^ii) («l I^^J I- «l l^i^«)] '
Now x^ |«i«9 = (ah |a?a) (O^i ~(a?i \xi) |a:,;
hence (a?! |ajia?2 1 . iCi \XiXt) = {xi \Xi) (xiX^ I a?ia5a).
And (xi {xioe^ , \XiX^u) = — (a^i^Wi) (^i !^)«
Thus cos y = (^l^»)*(^^«^n)
/>
Therefore p cos ;^ = ip.
297. Normals. (1) The normal at the point x is N^lx^x^, the normal
at the point x + ix ia
Hence (NN ') = | (xixJ^iSx^) = (xiX^XiSx^).
Now &ihsa^Sd+^Pi^^, Bx2=^a:i^0 + x^<l>.
Therefore
Therefore in general normals at neighbouring points of the surface do not
intersect. But [cf. § 295 (1) equation (ii)] normals at neighbouring points ou
a line of curvature do intersect,
494 CUBVBS AND SURFACES. [CHAP. TIL
(2) If the 0 and ^ curves are lines of curvature, then by § 295 (6)
Hence (^'hP'^hi'^) = 0 = (aifl?jflWPu)«
Thus (NIT) = (a?iawia^) B0B4>.
Hence neighbouring normals on the curve ds constant, or on the
curve (f> = constant, intersect ; that is to say, neighbouring normals on a line
of curvature intersect.
298. CiTRViLiNEAR Co-ORDINATES. (1) Let o; be conceived as a
function of three variables, 0, if}, ^fr. Then the equations d = constant,
^s= constant, and '^s constant, determine three families of surfaces. On
the surface, 0 = constant, a; is a function of the variables 4> ^i^d '^ ; on the
surfEice, ^=con8tant, it is a function of -^ and 0; on the surface, -^ = constant,
a function of 0 and ^.
Let o|g = ^i 5T=*^» S^fc."^" ^^^ * corresponding notation for the
higher differential coefficient&
(2) Now suppose that the three families of surfaces intersect orthogonally
wherever they meet.
Then (a?^ laj^) = 0 = (a?, |a?,) = (a^ |a?i).
Hence (fl?u|a^) + (flik«) = 0, («?i,|aJb) + (a2|^) = 0, (a^k) + (a^kn) = 0.
Therefore (oi f «») = 0 = (a, 1 w^) = (a^ i (HhaX
Also [c£ § 293 (2)] (a? k) = {w k) = (a \x,) = 0.
Hence since (xj \x)^0 — (xi\aJt) = (oBi\o^)^ {iVi [x^), it follows that
But the equations (a;, k) = 0 = (xx^^^) are the conditions [c£ § 295 (4)]
that the ^ and yft curves should be lines of curvature on the suiface, 0 = con-
stant. Thus the lines of intersection are lines of curvature on each surface.
This is Dupin's Theorem.
299. Limit-Surfaces. As a simple illustration of some of the above
formulsB, adapted to Hyperbolic Space, consider the limit-surface
€»(a?|a?) = (a?|6)».
It has been proved [c£ § 261 (3)] that if the spatial origin e be taken on
the surface, and if the line e^ be taken to be through the point 6 on the
absolute, then the equation takes the form
{x\x)^{x\{e + e,)Y.
Now if X be at unit intensity, we may write
298, 299] LIMIT-SURFACES. 495
Then aj^ = _- = ^c + ^g4.-e„
Let So- be the element of arc between the points x and x + Sx, then
i— ^ =s — (ajj |a?i) So-,' — 2 («i liTa) S<r,8<r, — (a?a |a?,) So-,"
" 7- '
Hence So* = So-,« + 8<r,«.
Accordingly the metrical properties of the surface must be the same as
those of a Euclidean plane. The same result had been arrived at before
[cf. § 262 (6)] when it was proved, that the sum of the angles of a triangle formed
by great circles on a sphere of infinite radius is equal to two right-angles.
The curvature (-] of any normal section [c£ § 294(1)] is given by
^ V{~ oh^ \(CiX^] ' {- (flg k) 8<ra« - 2 (ah l^^ySffjS^s^ (^ K) 8<r,«} _
^ ' {xXiX^^haf-^'^^xx^x^^hcjiat'V^xXiX^c^ha^
Hence every normal section is a limit-line, a result otherwise evident.
CHAPTER VIII.
Transition to Parabolic Geometry.
300. Parabouc Geometry. (1) The interest of Parabolic Geometry
centres in the fact that it includes the three dimensional space of ordinary
experience. Any generalization of our space conceptions, which does not at
the same time generalize them into the more perfect forms of Hyperbolic or
Elliptic Geometry, is of comparatively slight interest. We will therefore
confine our investigations of Parabolic Geometry to space of three dimensions,
in other words, to ordinary Euclidean space.
(2) The absolute quadric as represented by the point-equation has
degenerated into the two coincident planes [cf § 812]
(«!? 1 + ««? . + «.fa + ckS^y == 0.
The intensity of any point x{^ Sfc) must therefore [cf. § 213] be con-
ceived to be the square root of the left-hand side of this equation, that is,
ai^i + . .. + cla^a' The absolute plane itself being the locus of zero intensity.
(3) It is proved in § 87 that, if the intensities of the unit reference
points be properly chosen, the equation of the absolute plane becomes
f + f»+f. + f4 = 0,
and the intensity of any point 2{^ is Sf.
The intensities of all points in this plane are zero. Hence, if a and 6 be
any two points at unit intensity, the point a — b, which is at zero intensity,
lies in the absolute plane.
(4) If three of the reference points, namely, Ui, ti,, u^, be taken to be in
the absolute plane, and e be any other reference point, then any point x is
denoted by ^ + ^iih + ^fU^ + ^t^ ; ^^^^ i^ intensity is f . Thus the expression
e + Sfu is the typical form for all points at unit intensity.
301. Plane Equation of the Absolute. (1) In order to discuss
completely the formulsB for the measurement of distances and angles, it is
requisite to write down the most general plane-equation of the absolute,
which is consistent with the point-equation reducing to two coincident
planes. This question was discussed in § 84 (4).
301] PLANE EQUATION OF THE ABSOLUTE. 497
(2) Let any planar element be denoted by
Then it has been proved in § 84 (4) that the plane-equation
ctijXi' + . . . + 2ctuX^Xs + . . . ^0,
where the terms involving X are omitted, necessarily implies the point
equation,
P = 0,
where any point co is written ^e + f it^i + f jIa, + f ,14,,
The fully determined absolute quadric may therefore be considered as a
conic section lying in the absolute plane. The points on the absolute are
the points of the plane, the planes enveloping the absolute are the planes
touching the conic section. The absolute plane is also called the plane at
infinity ; and the conic section denoted by the plane-equation of the absolute
may be called the absolute conic lying in the plane at infinity.
(3) Let this conic section be assumed to be imaginary, so that the
elliptic measure of separation holds for planes [cf. § 211 (2)].
(4) It may be as well at this point to note that the operation of taking
the supplement with respect to the absolute becomes entirely nugatory.
The operation therefore symbolized by | will in Parabolic Geometry represent
as at its first introduction in § 99 the fact that the reference points (what-
ever four points they may be) are replaced according to the following scheme,
This operation of taking the supplement, as thus defined, will (as
previously) be useful in exhibiting the duality of the formulae, when it
exists. Its utility for metrical relations will be considered later [cf Book VII.,
Chapter li.]
(5) It has been proved in § 212 that in either Elliptic or Hyperbolic
space if we start with an absolute of the form,
and make it gradually degenerate to ^ = 0, at the same time increasing the
space-constant, then the distance between any two points x and y, where x
is f « + f 1^1 + f 2tt« + f it/j and y is i;e + i/ii^i + rnu^ + 17,1*, takes the form
^;
It will be observed that the assumption of the initial form of the absolute,
from which the degeneration takes place, is equivalent to the assumption that
eui, ett^y eiii are mutually at right-angles.
The most general assumption for the plane-ec^uation of the absolute Ls
then [cf § 84 (4)]
(ff^Jixy = /8,«Xi» + A»X,« + A%» = 0.
w. 32
498 TBAXRTIOV 10 PARABOUC CnOXBTBT. [CHAP. YUL
And if <f be the angle between the two phneB
and X'uii^tc,— V^<Mb + V^«i«i — V<^«s«
then [cf. 5 211] --^-.r^J^^y^lrry
(6) Bat the f 0 are not independent ci the /Pa aa theae two detadied
forma of statement may aoggeat. In oider to peroeiYe the oooneetion it
ia better to ooodact the gradoal degeneration oi the afaaohite aa fioUowa
Let the pfame-eqaation of the abaolate be
where fi will ultimately be made to Taniah.
Then the point-equation of the aheolate ia
Hence by reference to § 212 we see that
Therefore if IT be some finite constant,
Acoordingly, when /9 is made to vanish, the distance between two pmnts
X and y takes the fimn
K V(/3.«A* (g.iy - ^hfy + /8.'ft* (g,iy - ihgy + AW (f,if - fay}
If X and y be two points of unit intensity, they are of the form e + ^(v
and e + Si/tc, and their distance is
E Vl)S,W (f I - ih^ + AW (f . - i7.y + AW (fa - I7.f}.
902. Intensities. (1) The intensities of the points which lie on the
plane at infinity, which is the degenerate form represented by the point-
equation of the absolute, are all zero according to the general law of inten-
sity. It was explained in the chapter on Intensity [c£ § 86 (2)] that some
special law of intennty, applying to these points on the locus of zero intensity
must be introduced.
(2) Consider two points x and of on the line ^1. Let d; = tf-h{ii«if
of =^ e •¥ X^iUif so that x and of are at unit intensity. The distance ex is
Kfi^^u the distance eaf is Kfi^^i. Hence eaf = Xex.
302] INTENSITIES. 499
The differenoe of the two points x and e each at unit intensity is a point
at infinity, in &ct x^e^fiUi; similarly af --e^Xfiiii. Hence w^e and
of ^e denote the same point at infinity, but at intensities (according to some
new law) which are proportional to the distances ex and eaf.
(3) Let the intensity of a point at infinity be so defined that, if a and b
be any two unit points at unit distance, the point a — 6 is at unit intensity,
positive or negative. Also let the three points t^, t^, tisi VLsed above, be at
unit intensity.
Then any point a = 6 + t^i is a unit point on eui at unit distance from e.
But its distance from e is KPfit, Hence Kfi^fi^- 1* Similarly for points on
eUi and 6Us*
Thus /3i = /3,« A = )8, say ; and Kff*^ 1.
(4) Hence with these definitions, the plane-equation of the absolute is,
The angle 0 between the two planes
IS given by
cos OF as
vk V + V + V) (V + V' + vol '
The distance between any two unit points e + Sf u and e + X17U is
vKfi - v.r + (fa - v.r + (& - %)*}.
The intensity of the first of the planar elements given above is
V{V + V+V1.
The intensity of the point on the absolute plane, X,Ui + Xgtia + Xstt,, is the
distance between the points e and e + XXu, that is, V{V + X^' + X,*).
(5) The transition* from Hyperbolic or from Elliptic Geometry to that
of ordinary Space has now been fully investigated The logical results of
the definitions, which have finally been attained, will be investigated in the
next book.
* Since Enolidean spoee is the limit both of Elliptic and of Hyperbolic space with infinitely
large space-constants, it foUows that the properties of figoies in Elliptic or Hyperbolic Space,
contained within a sphere of radius small compared to the space-constant, become ultimately
those of flguxes in Endidean space. Hence the experience of our senses, which can never attain
to measurements of absolute accuracy^ although competent to determine that the space-constant
of the space of ordinary experience is greater than some large yalue, yet cannot, from the nature
of the case, prove that this space is absolutely Eudidean.
32—2
500 TRANSITION TO PARABOLIC GEOMETRY. [CHAP. VIII.
30B. Congruent Transformations. (1) It will however be instruc-
tive to work out the properties of CongrueDt Transformations for Parabolic
Geometry in the same way as that in which they were discussed in the
preceding chapter for Elliptic and Hyperbolic Geometry.
(2) The special properties of a congruent transformation are, as stated
in § 268 (1), (a) the internal measure relations of any figure are unaltered bj
the transformation : (13) the transformation can be conceived as the result of
another congruent transformation p times repeated, where p is any integer:
(7) real points are transformed into real points : (S) the intensities of points
are unaltered by transformation.
(3) It follows from (a) firstly that the plane at infinity is unaltered by
the transformation ; and secondly, that the degenerate quadric represented
by the plane-equation of the absolute, which is a conic in the plane at
infinity, is transformed into itself
(4) Thus the plane at infinity is one semi-latent plane of a congruent
transformation. It is proved in the next subsection that there must be at
least three distinct latent points on this plana Now, by reference to § 190,
it can be verified that semi-latent planes, with at least three distinct latent
points on them, only exist in the cases enumerated in § 190 (1), in § 190 (2),
in § 190 (3) Cases I. and II., in § 190 (4) Cases I. and II., and in § 190 (5)
Cases I. and II. But in each of these cases a semi-latent (or latent) line
exists, which does not lie in the semi-latent plane containing the three
distinct latent points. Now by Klein's Theorem [cf. § 200] the points on
the absolute on this line are the latent points of the congruent transforma-
tion. But these points on the absolute are the two coincident points in
which the line meets the plane at infinity. Hence the line is in general a
semi-latent line with only one latent point on it, namely, the point at
infinity.
(5) Now consider three unit points (ui, i/,, t^) on the plane at infinity,
so that the three lines drawn to them from a unit point e, not on this plane,
are at right-angles to each other. Then any point on the absolute is
f 1^1 + fi^ + f «^> a^d any plane is Xiijt^w, — Xjew^ti, + X^eUiti, — X,et/|U,. The
plane-equation of the degenerate absolute conic is V + X2' + X,^ = 0.
Hence, confining attention to points and lines on the absolute, any line
on the absolute is Xit^ti, + \9ihU1 + X^tiiu^' the line-equation of the absolute
conic is Xj" + X,* + X,* = 0 ; and its point-equation is f 1* + f g" + f,' =s 0.
Now it is easily proved* that a linear transformation in two dimensions,
which transforms a conic into itself, must be such that two of its latent
♦ Cf. Klein, lor, eit, p. 3C9.
303] CONGRUENT TRANSFORBIATIONS. 501
points are on the conic, and a third is the pole of the line joining the other
two.
Assume i^ to be the latent point not on the absolute conic: then the
polar of 1^ is the line tiiUz. Let this line cut the absolute in the points v and
v'. Then t^, v, t/ are the latent points of the transformation. Since the
conic is imaginary, the points v and t/ are conjugate imaginary points ; and
hence it is easily proved, that the three points v, v and t^ are necessarily
distinct. The equation of the line t/gtis is fi =» 0 ; hence v and v are given
by this equation and by f ,' + f ,* = 0. Thus we may write
(6) Let the latent roots of the matrix be a, fi, ff corresponding to
t^i, v, xf. Then fi and ^' must be conjugate imaginaries ; accordingly put
Again, considering the complete three dimensional transformation, the
semi-latent line, not lying in the plane at infinity, corresponds to two equal
roots. This repeated root must be real : hence the line also must be real,
and cut the plane at infinity in a real latent point. Thus u^ is the point in
which the semi-latent line cuts the plane at infinity. Now if p be any
point on this semi-latent line, and ^ be the matrix representing the complete
three dimensional transformation,
But from assumption (£) of subsection (2), a = 1.
Again*, in order that the conic may be transformed into itself, o^^ Pff,
Hence /3o^» 1, and therefore /3o= 1-
Thus finally the latent roots of the transformation are 1, 1, e^ and e~^.
(7) Now let t«3 and v^ be transformed into u^ and u^.
Then 6*<+ e" *<=6«t; = 6*^'''*)u, + 6"'^'"*)u„
Hence t*,' = m, cos S + t«i sin S, ii,' = u, cos 8 - t*j sin S.
Also let e be any unit point on the semi-latent line cutting the absolute
in t^. Then [cf. § 200 (2)] any point 6 + fwi on this line is transformed to
6 + (f + 7)t^. Thus all points on this line are displaced through the same
distance 7. Let this line be called the axis of the transformation.
Any point e + X^ becomes
« + (f 1 + 7) ^ + (f a cos S - f , sin 8) M, + (f , cos S + f a sin 8) «, .
* Cf. Klein, loe, cit, p. 869.
/
/
502 TBANSinON TO PARABOLIC OEOMISTRT [CHAP. VIIL
(8) If 2 = 0, the transfonnatioii is called a translation. The axis of a
translation is indeterminate, since any line parallel to &U| possesses the same
properties with regard to it as eu^.
If fy = 0, the transformation is a rotation* Every point on the axis eu^ of
the rotation is a latent point.
If any point at a finite distance is unchanged by a congruent transforma-
tion, then the axis must pass through that point, and 7^0. Hence the
transformation is a rotation.
i
I
I
I
1
BOOK VII.
APPLICATION OF THE CALCULUS OF EXTENSION
TO GEOMETRY.
CHAPTER I.
Vbctors.
304. Introductory. (1) The analytical formulae applicable to
Euclidean space relations were arrived at, under the name of Parabolic
Geometry, as a special limiting case of a generalized theory of distance.
We will now start afresh, and, apart from any generalized theory, will con-
sider the applicability of the Calculus of Extension to the investigation of
Euclidean Geometry of three dimensions. Neither will it be endeavoured to
assume a minimum of axioms and definitions in Geometry, and thence to
build up the whole science by the aid of the Calculus. Such a scientific
point of view was adopted in the investigation of the generalized metrical
theory of the previous book. At present the propositions of elementary
analytical Geometry will be assumed as known, and the suitability of the
Calculus for geometrical investigation demonstrated by their aid. It may be
further noticed that the propositions, which fall under the head of what is
ordinarily called Projective Geometiy, have been sufficiently exemplified in
Book III., so that now metrical propositions will be chiefly attended to.
Fia. 1.
(2) Let the points ei, 6,, e^, e^ form a tetrahedron; and let x be
any other point. Let the co-ordinates of a; in tetrahedral co-ordinates be
fii fj> f»» f* referred to the fundamental tetrahedron 616^6^4; so that, for
506 VECTORS. [chap, l
instance, {i is the ratio of the volume of the tetrahedron xe^R^^ to that of
the fundamental tetrahedron. Similarly for ^,, ^s, and ^4. Also fi is positive
when a; is on the same side of the plane ^^^4 as the point Ci, and ^1 is
negative when ^ is on the other side, with similar conventions for the signs
of the other co-ordinates. With these conventions the co-ordinates of x
always satisfy the equation,
(3) Now let Ci, e^, Bz, e^ also stand for four reference elements of the
first order [cf. §§ 20 and 94] in the calculus, and let w denote the element
fi^ + fjea + fi^ + f4^4- And let x be at unit intensity [c£ § 87], when
fi+ ft + fsH- ^4 = 1 ; and be at intensity X, when fi+ ?» + ?«+ f4 = X.
(4) Then, when iv is at unit intensity, the co-ordinates fi, |s, ^„ ^4 of
subsections (2) and (3) can be identified. For [cf. §§ 64 and 65] if a; andy
be any two points with tetrahedral co-ordinates fi, f,, f„ ^4 and 171, i/j, i;„ 1/4
respectively, then the point z which divides the line xy in the ratio X:/i,
so that iiisto^asXisto^, has as its co-ordinates
(Mfi + Xi;0/(^+M), (Mf. + >'W/(>'+/*), (Mf, + \i?,)/(X + Ai), (A*f4+Xi74)/(X+/A).
Thus if w and y also stand for unit elements in the calculus, the point z
stands for the element (am? + Xy)/(X + /a), and is also at unit intensity as thus
represented.
Thus conversely (jix + Xy) can be made to. represent any point on the
straight line osy, by a proper choice of X/fi.
(5) For instance let x, y, z denote the three angular points of a triangle
at unit intensity. The middle points of the sides, also at unit intensity, are
i (y + z), \(z-\- x), i (« + y). Any points, not necessarily at unit intensity,
on the 'three medians respectively are
iX(y + £r) + /ia?, ^\'(z + x) + fi'y, ^\"{x-{-y) + fif'z.
It is obvious therefore that the three medians meet in the point
(^ + y + z), which, as thus represented, is at intensity 3.
306. Points at Infinity. (1) The point fAx - Xy, assuming X and ^
to be positive, divides the line xy externally in the ratio X to ^ In
particular, the point x — y divides xy externally in a ratio of equality, and
is therefore the point on a;y at an infinite distance. It is to be noticed that
the intensity of ^ — y is necessarily zero, and therefore that the intensity
of \(x^y) is also zero. Thus the plane at infinity is the locus of points
at ^ero intensity [cf. § 86 (1)].
(2) A special law of intensity must therefore be assumed to hold for
the points on the plane at infinity [cf. § 86 (2)]. Thus i£ x,y,z be three
collinear points at unit intensity, y — x and z-^x both denote the same
point at infinity, but not at the same intensity according to this special law.
il
i
I
305, 806] POINTS AT INFINITY. 507
Suppose for instance that z divides the distance between x and y in
the ratio \ to /a, so that
s^(jiX'\-\y)l(\ + fi). Then -? - a? » -— - (y - a?).
Hence the intensity of z^ a; is to that of y — a; in the ratio of the distance
xz to that of the distance xy [cf. § 302 (2)].
(3) Any law of intensity may be assumed to hold in the plane at
infinity, which preserves this property [cf. § 85 (2)]. But great simplicity
is gained by defining the distance xy as the intensity of the element y-^.x,
(4) Let the line ab be parallel to the line cd and of the same length.
Fio. 3.
Let the points a, b, c,dhe at unit intensity. Then the elements 6 — a and
d — c are the same point at infinity at the same intensity.
Hence b^a^sd-^a, Therefore a— c«=6 — d, and the symbols express
the &ct, that ac and bd are equal and parallel. Also a + d^b-^c; which
expresses the fact, that cut and be bisect each other.
306. Vectors. (1) Let a point at infinity be called a vector line,
or shortly, a vector. A vector may be conceived as a directed length
associated with any one of the series of parallel lines in its direction.
Thus if u denote the vector parallel to ab and cd, and of length equal to
ab or cd reckoned from a to 6 or fix)m o to c2, then i^=:& — as(2 — c.
(2) The conception of vectors is rendered clearer by the introduction of
the idea of steps, which is explained in § 18. Thus the addition of t^ to a
is the step by which we pass fix)m a to 6, for a + u^b; and the intensity of
u measures the length of the step. Since also o + 1£ » c2, we must reckon,
in accordance with this definition, all parallel steps in the same sense and
of the same length as equivalent [c£ § 3].
Again if v denote the vector, or step, from d to 6, then 9 = 6 — d So
ii + v«c{ — C+&— ^=^6 — c. Thus the sum of two steps is found by the
parallelogram law.
508 VECTORS. [chap. L
(3) The fundamental tetrahedron may be chosen to have for its oomeis
any unit point e and three independent vectors Ui^u^.u^y each of unit length.
Any point x is then symbolized by {6 + {it^-f fatia + fstz,, and the intensity
of a; is f [cf. § 87 (4)]. Thus if a; be at unit intensity, it is written
^ + fit^+fst^+ fs^* Thus the lines euy, eu^, ev^ are three Cartesian axes,
And {i> fsi fs ^^ tihe Cartesian co-ordinates of the point. For let ei, e^, ^
be three unit points on the lines et^, eu^y ev^ respectively, and each at
unit distance fix)m e. Then t^»^ — ^, lUi^et — e, u^^e^^e. Also let
a? = f6 + fi6i + fjea + fj^> where f, fi, fj, f, are tetrahedral co-ordinates
of X. Then
But fi is the ratio of the tetrahedron exe^ to the tetrahedron eeie^, that
is, the ratio of that Cartesian co-ordinate of a, measured on ei^i, to a unit
length. Similarly for f, and f,. Hence fi, fs, f, may be considered as the
Cartesian co-ordinates of w, referred to the axes eui, eu^, eth.
(4) Any vector can be written in the form
A vector of the form \u + fiv must denote a vector parallel to the series
of planes which are parallel to the pair of vectors u and t;.
307. LiNEAB Elements. (1) A linear element, or the product of two
points, must be conceived as a magnitude associated with a definite line.
Thus, if a and b be two points, the linear element a6 is a magnitude asso-
ciated with the definite line ab.
Suppose that o is another point on ab such that the length from a to o is
X times the length from a to 6. Then dCs^Xab, c6 = (1 — X) a6. Therefore
c == (1 — X) a + X&, and c is at unit intensity, if a and b are also supposed to
be at unit intensity. But dc = Xa&. Hence the intensity of oc is X times
that of ab, when the length oc is X times the length ab,
(2) We may therefore define the intensity of the product of two unit
points as the length of the line joining them.
If the two points a and b are at intensities a and 13, the intensity ot abi&
afi times the length ab.
(3) The vector b—a(a and b being at unit intensities) and the product
ab should be carefully compared.
The intensity of each is defined as the length ab, but they are magni-
tudes of different kinds. For 6 — a is a directed length associated with any
one of the infinite set of straight lines parallel to ab, and is an extensive
magnitude of the first order, being really a point at infinity. While ab
307, 308] LINEAR ELEMENTS. 509
must be conceived as a directed length associated with the one definite line
ab, and is an extensive magnitude of the second order.
(4) Also ab can be written in the fonn a(b — a), since oa = 0. Hence
the linear element ab may be conceived as the vector 6 — a, fixed down or
anchored to a particular line; and the unit point a, as a factor, may be
conceived as not affecting the intensity, but as representing the operation
of fixing the vector.
(5) Also if c be any other unit point on the line ab, then
(a-c)(6-a) = 0;
since a^c and b — a represent the same point at infinity at different
intensities. Hence a(b''a)^{C'\'(a — c)] (6 — a) = c (6 — a). Thus any
other unit point in the line ab may be substituted as a factor in place of a.
(6) Hence if a, 6, c be three coUinear unit points,
a& + be « oc.
For by (5) 6 (c — 6) = a (c — 6) ; and hence
at + 6c = a (6 — a) + 6 (c — 6) = a (6 — a) + a (c — i) = a (c — a) = oc.
If a, by c be not collinear, then
where d is the opposite comer of the parallelogram found by completing the
parallelogram ab, ac,
308. Vector Areas. (1) A product of two vectors will be called a
vector area.
If uv be any vector area, then only the intensity is altered when any
two vectors parallel to the system of parallel planes defined by u and v are
substituted for u and t;.
For let Ui = X^u + /*,», ti, = \u + fi^v ;
then uiUi = {\fh -^ \fh) ^t^-
Hence u{U2 denotes the same vector area as uv only at different intensity.
510
VECTORa
[chap, l
(2) From any point e draw two linee ep and eq representing in
magnitude and direction the vectors u and v respectively, and complete the
parallelogram eprq. Also draw epi and ep^ to represent the vectors u, and «,
respectively, and complete the parallelogram ep^^.
Then, conceiving eu and ev as two Cartesian axes and assuming that
the vectors u and v are of equal length, the co-ordinates of pi and p^ are
\, /ii and X,, /i, respectively.
Hence the area of the parallelogram ep^p^ is to that of the parallelograin
eprq as (Xi/i^ — X^/ii) is to unity. Therefore the intensities otuv and UiUf are
in the ratio of the areas of the parallelograms formed by uv and y^tit.
(3) But the intensities of vector areas are necessarily zero according
to the general definition [cf. § 307 (2)] of the intensity of a linear element
For if au be a linear element where tt is a vector of length S and a is a point
at intensity a, then the intensity of au is a& But when a becomes a vector,
a is zero. Therefore the intensity of a product of two vectors is necessarily
zero.
Accordingly a special definition must be adopted for the intensity of
vector areas; and the above investigation shews that we may consistently
adopt the definition that the intensity is the area of the parallelogram
formed by completing the parallelogram eu, ev.
The intensity of uv will be considered by convention as positive when
by traversing the perimeter of the parallelogram so as first to move in the
direction of u and then of v, the direction of motion is clockwise relatively to
the enclosed area.
«
Then in the above figure for uv, we start fix)m e and traverse ep which
represents u and then pr which represents v, and the motion is anti-clockwise
so that the area is negative.
309]
VECTOR ARBAS.
511
(4) A vector area will be conceived as possessing an aspect or direction,
namely the aspect of the system of parallel planes which are parallel to the
two vectors. A line parallel to this system of planes will be called parallel
to the plane of the area, or parallel to its aspect.
309. Vector Areas as Carriers. (1) The addition of a vector area
to any linear element, which is parallel to its aspect, simply transfers the
linear element to a parallel line without altering its intensity.
Pio. 6.
For oft = a (6 — a) = {c + (a - c)} (6 — a) = c (6 — a) + (a - c) (6 - a).
Now let d — C'=^b^a.
Then c (d — c) -^{a — c){d — c) ^ a (b -- a).
Thus the addition to cd of the vector area (a — c) (d — c) transfers it to
ab, which is an equal and parallel linear element.
(2) It is also to be noticed that, if cd is conceived as continuously moved
into its new position by being kept parallel to itself with its ends on ca
and db, then it sweeps through the area ahdc, which is the area of the
parallelogram representing the intensity of the vector area.
(3) Let X and y denote any two unit points
«+fi^ + f2^ + f«w», and tf + i;it*i +i;,tia + i;,t«,.
Then by multiplication
This is the form which any linear element must assume. Any vector area
takes the less general form
512
VECTORS.
[chap, l
310. Planar Elements. (1) A planar element, or the product of
three points, must be conceived as a magnitude associated with a definite
plane.
Thus if abc be a planar element formed by the product of the thn^
points a, 6, and c, then abc is a magnitude associated with the definite
plane abc.
(2) Let Vri and t/, be two unit vectors parallel to this plane but not
parallel to one another, and let e be any other point in it. Then we may write
i«j.
Hence a6c= 111 ew,w
fli A 7i
Also eiiiUi = c (tf + t*i) (tf + ^^^).
Therefore the intensity of the planar element abc is to that of the planar
element e(e+ th) (e + u,) in the ratio of the area of the triangle formed bj
a, 6, c to that of the triangle formed by e, e + 1*1, « + ti,.
(3) We may therefore consistently define the intensity of the planar
element abc as twice the area of the triangle abc. Also the convention will
be made that the intensity is positive when the order of letters in (ibc directs
that the perimeter of the triangle be traversed in a clockwise direction.
If a be at intensity a, b at intensity fi, c at intensity y, then the intensity
abc is 2a)97 times the area of the triangle abc.
(4) In comparing a vector area with a planar element it must be noticed
that a vector area is conceived as an area associated with any one of a series
of parallel planes, while a planar element is conceived as an area associated
with a definite plane.
I
^
310 — 312] PLANAR ELEMENTS. 513
The planar element ahCy where a, b, c are unit points, can be written
in the form a (6 — a) (c — a). Then (6 — a)(C'- a) is a vector area of
which the area representing the intensity is the same in magnitude and
sign as the area representing the intensity of the planar element abc.
Accordingly if U represent a vector area, and a be any unit point, then
the planar element aU may be conceived as the tying down of the vector
area to the particular plane of the parallel system which passes through a.
Also this operation of fixing the vector area makes no change in the
intensity.
311. Vector Volumes. (1) A product of three vectors will be called
a vector volume.
The intensity of a vector volume is necessarily zero according to the
general definition of the intensity of a planar element. For if U he any
vector area of area B and a any point of intensity a, then the intensity of
aU is aS. Accordingly, when a becomes a vector and a is therefore zero,
the intensity of the planar element vanishes.
A special definition of the intensity of a vector volume must therefore
be adopted
(2) We may first notice that all vector volumes are simply numerical
multiples of any assigned vector volume. For let u^, v^, ttg be any three
non-coplanar vectors. Then since there can only be three independent
vectors, any other vectors u, v, w can be written respectively in the forms
Then by multiplication
uvw =
l^i^Wj.
Thus any vector volume is a numerical multiple of uitiitit.
(3) Also let two parallelepipeds be formed with lines representing
respectively z^, t^, tis and %v,w sls conterminous edges. Then the intensities
of Uit^st^ and uvw are in the ratio of the volumes of these parallelepipeds.
Thus we may consistently define the intensity of a vector volume as the
volume of the corresponding parallelepiped.
312. Vector Volumes as Carriers. (1) The addition of any vector
volume to a planar element transfers the planar element to a parallel plane
without altering its intensity.
w. 33
514>
VECTORS.
[chap. 1.
For consider any planar element abc and any vector volume V. Then
we may write
abc ==^a{b — a){C'- a),
and F=w(6 — a)(c-a),
where u is some vector.
Hence <ibc-^V = (a'^u)(b'~a){c-'a)
= a'(b'-a)(c'--a'),
where a' — a is the vector u, and a'b\ aV
are equal and parallel to ab and ac respec-
tively.
(2) Also it is obvious that if abc moves
continuously into its new position remaining
parallel to itself with its corners on aa\ bb\ ccf respectively, it sweeps out a
volume equal to half the volume of F.
313. Product of Four Points. (1) Since the complete region is of
three dimensions, the product of four points is a mere numerical quantity.
Let 6i, ^, ^, 04 be any four unit points forming a tetrahedron, and let
a, 6, c, (2 be any four other unit points, also expressible in the forms
%oey 'Hfie, Itye, "ZSe.
Fio. 7.
Then
(abcd)^
(eie^e^i).
fil> Aj A» a
7i» 7«» 7»» 74
Si, Sj, Si, 84
Accordingly (from a well-known proposition respecting tetrahedral co-
ordinates) the numbers expressed by the two products (abed) and (Cie^e^^)
are in the ratios of the tetrahedrons abed and 616^6^4.
(2) Let the product of four points, such as abed, be defined to be equal
to the volume of the parallelepiped, which has the three lines ab,a>c,adBs
conterminous edges.
Also [abed] = {a (6 — a) (c — a) (d — a)}.
Hence [abed] =:(aV), where F is a vector volume of volume equal to the
volume (abed).
814. Point and Vector Factors. (1) It has now been proved that
every non-vector product of an order higher than the first may be conceived
as consisting of two parts, the point factor, which will be conceived as of unit
intensity, and the vector factor. Also the intensity of the product, which
is either a length, or an area, or a volume, is also the intensity of the vector
factor.
"^
313 — 315] POINT AND vBcrroR factors. 515
(2) Thus any linear element can be written in the form au^ where u is
a vector line and a is a unit point ; any planar element ini the form aM,
where if is a vector area; any numerical product of four points in the
form (aV), where F is a vector volume.
(3) Also since a is a unit point and not a vector, it follows that au = 0
involves u = 0, and aM^ 0 involves Jf = 0, and aF= 0 involves F= 0.
Thus au = au', involves u = ^tf\
and aM = aM\ involves M = M';
and aV^aV\ involves F= V\
(4) Again, if a and 6 be two unit points in the line au, then au^hu.
If a and 6 be two unit points in the plane aM, then aM = hM,
If a and h be any two unit points whatever, then {aV)^{bV).
315. Interpretation of Formula. (1) It will serve as an illustra-
tion of the above discussion to observe the geometrical meanings of the
leading formulse of the Calculus of Extension in this application of it.
In the first place, let the complete region be a plane so that the multi-
plication of two lines is regressive [cf. § 100]. Let p, g, r, « be four points,
and let t be the point of intersection of the two lines pq and r«.
Fio. 8.
Then t^pq.ra ^(pqs)r- (pqr)8^ iprs) q - (qrs) p.
Hence t divides rs in the ratio of the area of the triangle rpq to that of the
triangle spq ; and the section is external, if the order of the letters in rpq
and spq makes the circuit of the triangles in the same direction ; and it is
internal, if the circuits are made in opposite directions.
Similarly t divides pq in the ratio of the area prs to qrs,
(2) In the second place, let three dimensional space form the complete
region. Then the products of a line and a plane, and also of two planes,
are regressive.
Let p, q, r,3,the any five points, and let si meet the plane pqr in w.
Then x = pqr,8t^ (pqrt) 8 - (pqrs) t = {pqst) r + {rpst) q + ( qrst) p.
83—2
616 VECTORS. [chap. 1.
Hence x divides 8t in the ratio of the volumes of the tetrahedrons j)^« and
pqrt ; and the section is external, if the products (pqra) and (pqrt) are of
the same sign, that is, if 8 and t are on the same side of the plane pqr:
otherwise the section is internal.
Also, the last form for x states that the areal co-ordinates of x refenred
to the triangle pyr are in the ratio of the volumes of the tetrahedrons grd^
rpnt,pqst.
(3) We may also notice here that according to these formulae any five
points in space are connected by the equation
(qrat) p — (prst) q + (pqst) r — {pqH) 8 + {pqra) < = 0.
The formulae for the line of intersection of two planes abc and de/sj^
abc . def= {abcf) de + {abce)fd + (abed) ef
= (adef) be + (hdef) ca + (ode/) ah.
The geometrical meanings of these formulae are obvious, though they
would be rather lengthy to describe.
m
316. Ybctor FoRMULifi. (1) Some of these formulae take a special
form, if four vectors t^, tXsi ^i, t^s be substituted for four of the points. The
special peculiarities arise from the fact that the product of four vectors is
necessarily zero ; and that if V be any vector volume, and a and 6 any two
unit points, then (aF) = (6F).
(2) The formula for five points becomes
(au^ViVi) Ml — (aiLiViV^) ii, + (auiU^v^) v, — (attiV'^i) v^ = 0.
(3) Again, auiV^ . v^v^ = {au^u^v^ Vj — {av^u^v^ Vj = {auiv^v^ m, — {aujOiV^ Ui,
Also at^iVa . UiU^ = (av^v^u^ u^ — {aViV^a^ v^ = {o^ViV^il^ v^ — (av^VriU^ Vi,
Hence auiV^ . ViV, + aviv^ . UiV^ = 0.
Or, if M and M' be any two vector areas,
aM.M'^aM'.M^O.
Again, it is obvious that, if a and 6 be any two unit points,
aM.M'^bM.M\
317. Operation of Taking the Vector. (1) Let a unit vector
volume be denoted by the symbol U, and let the sense of U be such that
(aU) = 1, where a is any point of unit intensity.
Also if tt be any vector volume, and if M be any vector area, then (t*U) = 0,
and (i/U) = 0.
(2) Now, if F be any linear element, it can be written in the form au
and F\l = au,Vi — (all) w = i/.
316, 317] OPERATION OF TAKING THE VECTOR. 517
Similarly if F denote any planar element, it can be written in the form
aM, andP.U = aJf.U = (aU).Jf=Jf.
Hence the operation of multiplying U on to any non-vector element
of any order yields the vector factor of that element. This operation will
therefore be called the operation of taking the vector.
(3) We may notice that, if this operation be applied to a vector, the
result is zero ; and if to a point, the result is the intensity of the point with
its proper sign.
(4) Thus if any force be ah where a and h are unit points, by taking
the vector we have by the ordinary rule of multiplication
a6.tt = (aU)6-(6U)a=6-a.
Again, if any plane area be abc where a, 6, and c are unit points, taking
the vector we have
afcc . tt = (aU) he + (6U) ca + (cU) a6 = 6c + ca + 06,
which is therefore the required vector factor.
(5) In considering the effect of this operation on regressive products,
it is well to notice that, if |) be any point, the product ( j>U) can be conceived
both as progressive and as regressive. Therefore the multiplication of U
on to any pure regressive product still leaves a pure regressive product,
which is therefore associative.
(6) The regressive product au . hM is a point, so taking its vector must
yield the intensity of the point. Also the product au.hM.W is a pure
regressive product and is therefore associative.
Hence au . hM . tt = ai^ (bM . U) = {auM).
Therefore {auM)^ which also equals (fmM\ is the intensity of the point.
(7) Again, aM . hM' is a linear element, and its vector factor is given by
aM .hM\U = aM,{hM\\X)^aM .M' ^-hM\M.
Also, since the result is a vector, it is evident that any unit point c can
be substituted for a or 6 in these two formulae for the vector factor. These
results should be compared with the last formulae in § 316 (3).
(8) Finally aM . hM' . cM" is a point. To find its intensity take the
vector, then
aM, hM' . cM" .W^aM.hM'. {cM" . U) = aJlf . hM' . M".
(9) As an illustration of these formulae, let us find the vector £Eu;tor of
abc . def. Then by subsection (4) of this article
abc . def. tt = abc . {def. U) = abc . {ef-\'fd + de)
= — def, (he -^ca-^- ah).
518 VBCTOES. [chap. I.
Also
abc . (e/+/d + de) = (abciT) (/- e) + (abce) {d -/) + (abcf) {e - d).
(10) Again, let a be any unit point, and F be any linear element Then
F can be written be, where b and c are unit points.
Hence ajP . tt = oic . U = (6c + ca + oft)
= ^+a(6-c).
But c-6 = jP. U.
Therefore aF.U^F-a.FM.
(11) Let ^ be any linear element. Then the linear element through
any point d parallel to jP is d . FVi, Thus if ^ be in the form ab, where a
and b are unit points, the parallel line through dis d{b'- a).
Let P be any planar element. Then the plane through any point d
parallel to P is d . PU. Thus if P be in the form abc, where a, b, and c are
unit points, the parallel plane through (2 is (2 (&c + ca + a&).
318. Theory of Forces. (1) The theory of forces or linear elements,
as discussed in Book V., holds in the Euclidean Space now under discussion.
But some further propositions involving vectors must be added.
(2) In § 160 (2) it is proved that, if a be any given pointy and A any
given planar element, then any system of forces 8 can be written in the form
8^ap + AP,
where p and P are respectively a point and planar element depending on the
system 8.
Now let A denote a vector volume ; then AP denotes some vector area,
call it M. Also ap can be written in the form au, where uisA vector.
Thus iS = aw + Jf.
(3) The vector u is independent of the point a. For taking the vector
of both sides
8Vl-=au.n + MU = au.Vi=^u.
Hence, since u can be written in the form SW, it is independent of any
special method of writing 8.
(4) Let 8Vi be called the * principal vector* of 8. It is the sum of the
vector parts of those separate forces which can be conceived as forming 8.
Let the vector area M be called the vector moment of the system round
the point a, or the couple of 8 with respect to a.
Let a be called the base point to which the system is reduced.
(5) Also M depends on the position of a. For a8 = aM.
Hence M=a8 .U, and therefore M is the vector factor of the planar
element aS, which is the planar-element representing the null plane with
respect to a.
I
V,
318] THEORY OF FORCES. 61S
The same results respectiDg M and u follow directly from § 317 (10). For
by adding the results of applying the formula of that subsection to ecu^h
component force of 8, we at once obtain
iS=ra.iStt + ofif.tt.
Let a! be any other unit point, and let M' be the vector moment of 8
with respect to it. Then
/S = at« + if={a' + (a-a')}M + if = a'w + {(a-aOw + -5'^l-
Hence M'^{a^ a') m + Jf.
(6) Also {88) = 2 (auM) ; and since uM is a vector volume,
(auM) = (a'uM) = (a'uMy
And since aM = a8, (88) = 2 (avS), where u is the principal vector of 8,
(7) Again evidently
(aa'M) = (aa'Jf ') = (aa'8).
And (au/g) = ^ (88) = (a'wfif).
Therefore {(a — a) u8} = 0, where a and a' are any two unit points. This
is only an expression of the fact that u8 is a vector volume, where u is the
principal vector of 8. In fact from § 167 (2) we have
u8^8.8U^^(88)\l
Thus the plane at infinity ia the null plane of the principal vector.
(8) To find the locus of base points with the same vector moment M.
Let a be one such point and x any other such point. Then by
hypothesis
Hence a?(flf — Jtf) = 0. But flf — if is the linear element a . fill. There-
fore the equation, x(8^M) = 0, denotes that a lies on a straight line parallel
to fill
(9) Let Mo be any given vector area, then if OM^ be the vector moment
of 8 (round an appropriate base point) which is parallel to M^, uS= 0uMo,
where u is the principal vector of 8, Hence if a be any unit point
(au8) = i (88) = 0 (auMo).
Therefore 0^1 ^^^
2 (auMoY
Thus the locus of a point x such that the vector moment of 8 with
respect to it is parallel to Jlf^— or in other words, the point of which the
null plane with respect to £> is parallel to Mo — ^is given by
'(«-i(S)*)=»-
520
VECTORS.
[chap. I.
But it was proved in § 162 (2) that the conjugate of any line ab is
1 (SS^
S — a / i.g\ ^^' H^i^ce the conjugate of any vector area Jf o is a straight
line parallel to the principal vector, and this line is also the locus of points
corresponding to which the vector moments are parallel to Ma.
319. Graphic Statics. (1) It will illustrate the methods of the
Calculus of Extension as applied to Euclidean Space, if we investigate at
this point the ordinary graphic construction for finding the resultant of any
number of forces lying in one plane.
(2) Let the given system, 8, of coplanar forces be also denoted by
OiWi + Osttiz + ... + at,u,; where Oitti, Oati,, etc. are given forces (cf. fig. 9). We
require to construct their resultant.
u
p+i
«P-i
Fio. 9.
Let V be any arbitrarily assumed vector in the plane.
ih = v + (v^-v),
t«2 = (t; - t^i) + (t^ + i«a - 1;),
Wj = (v - Wi - Wj) + (til 4- Wj + 1*, - v),
Thus
Then
1/^ = (V - l/j ... - Ufy^i) + (Wi + Ma + ... + W|,— V),
iSf = aiV -h (Oa — Oi) (t^ — Wi) + ...
+ (ttp — ^iH-i) (t^ — t*i ... — W^i) + ...
+ (a„ — a„-i) (t; — 1*1 ... — u^-i)
+ a^(i«i + t«2+ •••+«*•' — v)
.(i>
.(2).
319] GRAPHIC STATICS. 621
(3) The equations (1) giving the vector parts of the forces' are equiva-
lent to starting from any point &« (cf. fig. 10) and drawing &«c to represent v
and bobi to represent Wi. Then biC represents v — Ui. Also from bi draw 6162
Fio. 10.
to represent ti^, then b^fi represents t; — t^ — i/a, and so on. This is the
ordinary Graphic construction for the force polygon with any pole c.
(4) To simplify the expression for 8 notice that ai, Oa, ... a^ may be any
points on the lines of the forces ; hence we may assume (cf. figs. 9 and 10) that
Oa — Oi is drawn parallel to v — Ui, a, — Oa parallel to i; — t^ — Wj, and so on.
Then flf = c^v + a„ (ui + t^a 4- ... + w,r — v).
Thus the resultant force passes through the point d which is the point of
intersection of Oit; and a„ (tti + w, + . . . + 1^„ — v), and is the force
d(wi + t«a + ... +Wr).
(5) This gives the ordinary construction for a funicular polygon, thus :
start from any point a^, draw o^ parallel to v, OiOs parallel to v—Ui, and so
on, finally apO^+i parallel to t; — tti — 1^... — w„. Then the resultant passes
through the point of intersection of a^ and a^a^+i.
(6) Suppose that two different funicular polygons are drawn, namely
aoOi ... a^ and Oo'oi' ... a/, corresponding to the arbitrary assumptions of two
522
VECTORS.
[chap. I
different vectors v and t/ respectively with which to comnience the constnic-
tions (cf. fig. 11). We will prove the well-known theorem that the points of
intersection of corresponding sides are collinear.
For affibft' is parallel to u^, hence ap,Up==apUp.
Again, apfif^i is parallel to v — Ui — ..,— w^,, hence
ap(v — tti — ... — Wp_i) = ap_i (v — t^ — ... — tt^i).
Similarly, a/ (v' — i^j — . . . — w^i) = a'p»i (v —Ui — ...— w>-i).
Therefore aiV — OiV = Oi (t; — t^i) — a/ (t;' — i^)
= Oa (V — ^) — ««' (v' — t*,) = Oa (V — Wi — tia) — «a' (^ — ^ "" ^O
= a, (v — 1^ — tia) - a,' (v — 1*1 — Oj) = etc.
Let ap^ittp and a'^^a; intersect in d^„ then
OiV — Oi V = do (t^ — v), (hiv — Ui) — Oa' {v — t^) = di (v - v'),
and so on.
Hence d^(v — v') = di(v — v') = ... = dr(v — ^')-
Fia. 11.
Thus the points do, dj, etc. all lie on a straight line parallel to t; — v^
Also if the force polygon bjbib^ etc. be identical in the two cases (cf. fig. 11),
and if b^ represents v and b^fi' represents v\ then cc' is parallel to v — v'.
Note. Grassmann oonsiders vectorB in the Ausdehnungdehre von 1862 ; but not in
connection with points. The formuke and ideas of the present and the next chapter
are, I believe, in this respect new. The two operations of 'Taking the Vector' and of
* Taking the Flux * [cf. § 325] are, I believe, new operations which have never been d^Sned
bef<Mne. Since this note was in print I have seen the work of M. Burali-Forti, mentioned
in the Note on GroMmann at the end of this volume.
CHAPTER II.
Vectors {continved).
320. Supplements. (1) The theory of supplements and of inner
multiplication has important relations to vector properties.
Take any self-normal quadric [cf. §§ 110, 111], real or imaginary, and
let Ci, ^„ e^y €4 be four real unit points forming a self-conjugate tetrahedron
with respect to this quadric. Let fi, e^, €,, €4 be respectively the normal
intensities [cf. § 109 (3)] of these reference points, where 61, 69, e,, €4 are any
real or pure imaginary quantities. Then
Also if X be any point 2^6, then
Suppose that y, which is Xtfe^ is on the polar plane of x with respect to
the quadric, then
€1 Cj ©8 ^4
Also all the points normal to x with respect to the quadric must lie on
this plane.
(2) Hence the pole of the plane at infinity is the only point which has
three vectors, not coplanar, normal to it. This is the point
6l* €,» €,« €4* €i' + 6,* + €,*-h€4''
This point is the centre of the quadric. Let it be denoted by e, where e
is at unit intensity, and let the normal intensity of « be e. Then we may
write
Hence (ele) = ^- __-i-^-.
Therefore €» « ei" + Cj' + €,« + 64'.
524 VECTORS. [chap, il
(3) But 61, es, 6,, 64 are any four points forming a normal system with
respect to the quadric. Hence the last equation proves that, when the quadric
has not its centre at infinity (in which case €1' + eg' 4- €j' + €4' = 0), the sum
of the squares of the normal intensities of any normal system of points is
constant and is equal to the square of the normal intensity of the centre.
321. Rectangular Normal Systems. (1) Now let e be the centre
of the self-normal quadric, and let v^^ u^y u% be three unit vectors forming
with 6 a normal system with respect to the quadric. We may assume
without loss of generality that the normal intensity of « is unity. Since
^> tis, lis lie in the locus of zero intensity their normal intensities according
to the general definition of intensity holding for all points, are zero. But
[cf. §§ 109 (3) and 110 (1)] let the normal lengths (or intensities) of w^, «,, «,
be tti, Og, ocs, according to the special definition of intensity for vectors [cC
§ 305 (3)].
Ill
Then (e|e) = l, (t^|wi) = -i, («2|wa) = -i, (^K) = -^.
(2) Also any point x at unit intensity is of the form e + X^ Hence
(a?k) = l+^V^ + ^'.
Thus the self-normal quadric is,
and is accordingly purely imaginary, unless one or more of oti, Oa, a, are pure
imaginaries.
(3) It is obvious from the equation of the self-normal quadric that, if
Ux^u^yU^ be any three mutually normal vectors, then ^, eu^, eu, are three
conjugate diameters of the quadric.
In general one set and only one set of such conjugate diameters are
mutually at right angles. But if the quadric be a sphere with a real or
imaginary radius, then all such sets are at right angles. In such a case
let the normal systems be called rectangular normal systems. The centre (e)
of the self-normal sphere will be called the Origin.
322. Imaginary Self-Normal Sphere. (1) Firstly, let the sphere
be imaginarj' with radius V(— 1). Then aj = Oa = a, = 1.
Hence if tti, ti^, 2^ be any set of unit vectors at right angles, then
(Ui |t^) = 1 = {u^\u^ = (u, |tt,) ;
and (ua |t*s) = 0 = {v^ \u^ = (wi \u^.
Also if e be the centre of the sphere,
(e|wi) = 0 = (e|M2) = (6|i/8),
and (e|e) = l.
321-323] IMAGINARY SELF-NORMAL SPHERE. 525
Again, |6 = ti,ti,t*3, !tti = — 6w«Wa, |tia = — eu,Mi, |w, = — ewiti,.
And { etii = u^u^y \ u^v^ = eii^ ,
and \eu^ = Wsiti , |«,w, = eti,,
and {et£, = U1U2, \uiU^ = eti^.
Also I Wjl^atls = — C, |eiZ2t^s= t/i, |^a,tii =:tis, {6Ui1^ = t«s*
(2) Let V be any vector f it^ + f jUg + f jW,, then (v v) = fi' + fa'H- fs*.
Hence (1; jv) is the square of the length of v.
Again, let v and t/ be two vectors fitii + f,ti, + f jWj and fiX + f j'^a + fj'tij,
of lengths p and p' respectively. Then
(« If') = fif/ + f A' + f»f.' = PP' cos 0,
where 0 is the angle between the two vectors.
Thus co3g- ^"""^^ and sin ^- / <^'^>
(3) Again, let M be any vector area ^lU^Us + (2^'fh + ^3^^> c^d let /i be
its area.
Then (M\M) = f,» + f,« + f,» = /i«.
Also let Jf' be another vector area in a plane making an angle 0 with
that of M; let M' be written f/w,M, + fs'i/^Uj + fj'ttiiia, and let its intensity
be /i'.
Then (if |Jlf 0 = fif/ + f,f; + fa?.' = /i/ cos ft
(4) The inner squares and products of points and linear elements in
general have no important signification.
It will be observed that these formulas for the product of two vectors or
of two vector areas are entirely independent of the centre of the self-normal
quadric.
The expressions (Jf |ilf) and (u\u) will be often shortened into M* and
u\ on the understanding that the normal system is rectangular and the
radius of the self-normal sphere is V(— 1).
323. Real Self-Normal Sphere. (1) Secondly, let the self-normal
sphere be real with radius unity. Then with the notation of § 321
ai=at = at = V(-l).
(2) Hence if lii, ti^, u, be any set of unit vectors mutually at right-angles,
then
(tti |wx) = - 1 = (mj |m,) = (w, |w,) ;
and (t^ I M,) = 0 = (i^a \ui) = (u^ | m,).
Also, if 6 be the centre of the sphere,
(6|i^) = 0 = (6|wj) = (c|ti3);
and (e \e) = 1.
526 VECTORS. [chap, il
Thus if V and 1/ be any two vectors Sfu and %^'u, of lengths p and p\ and
making an angle 0 with each other,
(f K) = - (fif/ + f.f.' + f.f.') = - PP' cos <?.
And in fact a set of formulse can be deduced analogous to those which
obtain in the first case, when the sphere is imaginary with radius V(' 1).
(3) But the constant introduction of the negative sign is very incon-
venient, we will therefore for the future, unless it is otherwise expressly
stated, assume that rectangular normal systems are employed and that the
self- normal sphere is imaginary as in § 322.
324. Geometrical Formula (1) If k and v are vectors the square of
the length of w + » is given by
(u + vy=(u -f v) |(w + r) = u* + v» + 2 (w \v)
= w« + 1^+ 2 V(t*V)cos ^,
where 0 is the angle between u and v.
(2) To express the area of the triangle abc. Assume that a, 6 and c
are at unit intensities. Then taking the vector
abc . U = 5c + ca + 06 = (a — 6) (a — c).
Hence, if A be the required area,
A' = Ha6c.U)> = i{(a-6)(a-c)|(a-6)(a-c)}
If a, 13, y he the lengths of the sides, and a, jS, y the corresponding
angles, (a — 6)* = 7", (a — c)* = ^, (a — 6) |(a — c) = ^87 cos a.
Thus A» = i)8V8in»a.
(3) The angle, 0, between two linear elements Cj and C^ is given
[cf. § 322 (2)] by
The angle, 0, between two planar elements P and Q is given [c£ § 322 (3)]
by
a _ (Ptt IQU)
*^ " - vKPu)- mn '
(4) The length of the perpendicular from any point a on to the plane of
the planar element P is obviously [c£ §§311 (3) and 322 (3)]
'/(Pny
(5) To find the shortest distance between two lines (7| and Cg.
Let Oi and a, be any two unit points on Ci and C^ respectively. Then
Ci = Oi . CjU, and (7, = a, . CJX. Also the required perpendicular («j) is equal
to the perpendicular from a, on to the plane ai . (7iU . Calt.
324, 325] OSOMETRIGAL FORMULA. 527
(6) If 7i and y^ be the lengths of the linear elements Ot and 0^, 0 the
angle between them, and vr the length of the common perpendicular, then
[c£ § 822 (2)]
((7iU.CiU)' = 7,V8in*^-
Hence (CiC,) = 7i7i«r sin 0.
We may notice here that
8ing = Vll-co8«g} = ^ (fiUy(C,tt)» •
325. Taking the Flux. (1) It is often necessary to express the
vector-line normal to a vector-area, or the vector-area perpendicular to a
vector-line. This is accomplished by a combination of operations, which we
will call the operation of taking the ' Flux.' Let v be any vector
and let e be the origin. Then
Accordingly \ev is the vector-area perpendicular to the vector v. Also
the length of v is V{f i' + i% + fa'} times the unit of length, and the area of | ev
is V{fi'+ ^2* + iz] times the unit of area. The vector-area \ev will be called
the flux of the vector v.
Again, let M be the vector-area
Then | alf = f 1 1 eu^u^ + fa | ev^Uy + f , |eaiUa = f i^i + f a^ + f s^-
Hence the vector-line |eJlf, which has been defined as the flux of Jf, is
perpendicular to the plane of Jf.
The operation of taking the flux can of course be applied to non-vector
elements, but the results are of no interest as permanent formulsB and are
easily worked out afresh when required.
(2) It will be noticed that the result of the operation on vector elements
is really entirely independent of the position of e, the centre of the self-
normal sphere.
Also furthermore the operation is capable of an alternative form. For
|6v = !6|v = U|v = -|».U; and |6Jlf= |e|Jlf = U|-af = l-Jf . U.
Hence, except in respect to sign, the operation is identical with that of
taking the vector of the supplement of the vector-line or of the vector-area.
(3) For these two reasons it is desirable to adopt a single symbol for
the combination of operations denoted by |e. Let \ev and \eM be written %v
and %M respectively.
528 VECTORS. [chap, a
We notice that %%v = v and %%M = M.
Also the operation is distributive as regards addition ; but it is not
distributive as regards multiplication.
326. Flux Multiplication. (1) The operation of multiplying the
flux of a vector-line or vector-area into a vector-line or vector-area will
be called Flux Multiplication. Its formulae are almost identical with those
of Inner Multiplication [cf. § 119]. They are all independent of the position
of the origin : this fact will be obvious, if it be remembered that (eVL) = («'U),
where e and e' are any two unit points.
(2) Firstly, it is obvious that
{v%v) = (f ,» + f ,« + f ,0 « = (v \v) U,
(ilfgilf ) = (f ,« + f «« -h f ,«) U = (Jlf I ilf ) U.
Similarly (vgv') = (i; 1 1/) U = (t/gv),
and (mM') = (Jlf | ilf' ) U = (M'%M),
(3) Again, Vi^v^v^ = Vi (| v, . Iv, . U) = (vi | v,) U | y, + (vi Iw,) | v, . U
= (t'i|V8)gVa-(v, |v8)gv, (i).
And Vjt^sgv, = 0 = if gvi (ii).
Equation (i) may be compared with
Vi |VjVs = {Vi \Vi) I Va - (Vi |V,) !»,.
From equation (ii), it follows as a particular case that
%Vi . gVg = 0.
(4) From equation (i), replacing Vi by gi^w', we have
gW . gW = (gutt' . |t/) gv - {%uu' . I v) gt/.
But [cf. § 99 (4)] gjf \v = l(eifv) = (eMv).
Hence gtiw' . gtw' = (cawV) gw - (eww'v) gt/ (iii).
As a particular case of equation (iii) we deduce
gtw'.gtw" = (etn;V')gv (iv),
(5) Let a be any unit point, then
agu = - a (I u . U) = - (aU ) 1 u + (a I m) U = - 1 w -f- (a I ti) U.
Hence
aga . gv = - {|u - (a |u) U} gv = - |m . gv = |w . jv . U = I ttv. U = gw» (v).
Also if 6 be another unit point, and M and M' be vector-areas,
aJf.6gif' = (aJI/gJf05-(aM6)gif' = (Jlf|Jlf')6-(aJf&)gJlf' (vi).
And as particular cases
aM.b%M^M^b^{aMb)%M\
aM.a%M'^{M\M')a ] ^^"^*
'U
i
326 — 328] GEOMETRICAL FORMULiE. 529
327. Geometrical Formula. (1) To find the foot of the perpen-
dicular from the vertex d of the tetrahedron abed on to the opposite &ce.
The required point [cf. §§ 317 (4) and 325 (1)] ia abc.d%(bc + ca + abX
and by the transformation of equation (vii) this can be written
{bc + ca-^- ahf d — {abed) % (be + ca+ ah),
(2) To find the line perpendicular to two given lines.
Let Ci and C, be linear elements on the two given lines. Then Ci\X . CqU
is a vector-area perpendicular to the required line. Hence % (C,U . CjU)
is a vector parallel to the required line. But the required line intersects
both the lines Ci and (7,. Hence it lies in both the planes Ci% (Ci\X . 0,11)
and CaS (C,U . CjU). The required line is therefore
(3) Now if the two lines Ci and Ca are given respectively in the forms
Oit/i and Oai/a) this expression for the line of the common perpendicular can
be transformed by equation (i).
For CiB (CiU . CaU) = OiWigt^Ua = Oi . tk^UiUi
= K |t^) OlSWi - (W, \U,) OigWa,
and similarly Ogg (CJiX . CiU) = (t^i I w,) Oagwa - (t«, I w,) Oagi^ .
Hence the required line is now [cf. § 102 (7)] in the form
(t*i {U^y Oi^Ui . a,gWa+ (t*i |Wi) (Wa I Wa) O^gt^ . OagWi
- (t^ |Ua) (l£a |Wa) (OiOsgw,) g^i - (it, \u^) (l*i t^) (OiOagM,) gt^.
328. The Central Axis. (1) It follows as a particular case of
§ 318 (9) that any system S can be written in the form au + w%tc ; where
a is any point on a certain line parallel to u, called the Central Axis, and «r
is called the pitch. Also [cf. § 318 (9)]
-1 (^^l-i(^ = i(^
(2) It is obvious that S can be written, without the use of u, in the form
a . iSfU + «rg (flfU).
Also other expressions for «r are found from
(/S |fif) « tt« (1 + isr») = {Sny (1 + «r«).
Hence 1 +t«r»« ^^_ , and ^-^. = ^-^^ .
(3) We are now prepared to discuss the signification of the addition of
any vector-area M to any force system S,
w. 34
530 VECTORS. [chap. II.
It has been proved [c£ § 309] that the addition of JIf to any linear
element Gi in its plane is equivalent to the transference of Ci, kept parallel
to itself, to a new position, so that in the transference C^ sweeps out an area
s/iMy with the aspect of M,
Let 8 be written in the form au + m%u. Then if M and $u have the
vector w in common, M can be written in the form X^w + vm.
Thus 8'\-M — au + v^u + ls$u-\-wu
= (a + w) «^ + (bt + \) gt*.
Hence the component of M which is parallel to the central axis (namely,
wv) transfers the central axis according to the ordinary rule for transferring a
linear element parallel to the plane of the vector; and the component of if
which is perpendicular to the central axis (namely, \%u) simply alters the
pitch. It will be noticed that the principal vector of £f is unaltered.
329. Flaxes coxtainixq the Central Axis. (1) The plane through
the central axis of 8 and parallel to any vector v is
flf»-tir(t;|.flfU)U,
where w is the pitch of 8.
For, writing 8 in the form au -f wSw, the required plane is
auv = (iSf — -bjSw) t; = Sv — w%u = flft; — w (r 1 1^) U.
If v be perpendicular to /SU, the plane becomes 8v,
(2) Hence the plane containing the central axis of 8, and perpendicular
to the given vector area M, is
S^M^^{8M)\l
For by subsection (1) the plane is Sgif - isr (gif . \8Vi) U.
Now (gJlflflfU) = (|cif.|/SU) = (6Jf;/SU) = («ifU.flf) = (JfS);
since the three terms eMy 8 and U, form a pure regressive product [cf. 101 (3)].
330. Dual Groups of Systems of Forces. (1) The metrical pro-
perties of dual groups — and therefore also of quadruple groups [cf. §§ 169
and 170] — can now be discussed.
Let 8x and 8^ be any two systems defining a dual group. Let 8i be
written in the form a^Vi + -sti^Vi, and 8^ in the form a^v^-^-'sr^v^.
Then {8Si) = (^^1^102^2) + ^^i (fl^v^v^ + «•« (oa^aSvi)
= (aiViO^V^) + (tSTi + «ra) (Vi |Va).
Accordingly [cf. § 324 (6)] if tf be the angle between OiVi and a^t, and
B the shortest distance between them,
(iSjiSa) = Vl^'i V} . {d sin tf + (tsTj + tsTj) cos 0}.
329 — 332] DUAL GROUPS OF SYSTEMS OF FORCES. 531
/or cr\
(2) Let -w. „;..,,''- -.i-. be called the virtual coefficient of the two
systems 8i and 8^, This virtual coefficient can accordingly also be written
in the form {dmi0 + {vfi + «r,) cos 0]. The idea of the virtual coefficient,
and this latter form of it are due to Sir R. S. Ball in his Theory of Screws.
(3) Since the condition that the two systems be reciprocal can be
written
dsin ^ + (-BTi + t!rj,)cos ^= 0,
it follows that if the central axes of two reciprocal systems intersect, either
they are at right angles, or the sum of the pitches is zero.
331. Invariants of a Dual Group. (1) A vector-area parallel to
the two central axes is SiVi . /S9U. This vector-area is an invariant [cf. § 177]
of the system. For let
Then /SfU . flf' U = (X^/i, - X^) flf^U . fif,U.
Hence all the central axes of the dual group are perpendicular to the
same vector % (flfjU . fifjU).
(2) The plane through the central axis of 81, and parallel to the normal
to the vector-area S^U . flf,U is, by § 329 (2), S,% (8^Vi . iSf,U).
Hence the line of the shortest distance between the central axes of
81 and 8i is
Si% (8,n . flf,U) . 8,% (S,U . /S,U).
But this linear element is an invariant of the group. For substituting
8 and 8' for 81 and fifj, we find by the use of the previous subsection
S.g(fifU.flf'U).S'g(/8fU.fif'U)
= (Xi/i. - X^)» (XiSi + X^.) g (Sill . S«U) . Oi,Si +/!,«,) 8 (i^^^
= (Xi/^ - X^y 8,% (8M . 8^Vi) . fif.g {8M . 8,Vi).
Hence all the central axes of the systems of a dual group intersect the
same line at right angles. Let this line be called the axis of the group.
332. Secondary Axes of a Dual Group. (1) In any dual group
there are in general two, and only two, reciprocal sjnstems with their
central axes at right angles.
For let 8 and 8' be two such systema Then {88') = 0, gives
Xi/i. (SiflfO + X^ (SA) + (X,/*, + X^KSiiSf,) « 0 ;
and(i8fU|iS'U) = 0,give8
\fh (SiUy -I- X^ (S,U)» + (Xj/i, + X^) (8^U l&U) = 0.
84—2
582 VECTORS. [chap. n.
From these two equations by eliminating /<h//ii, we deduce
V l(S,flfO('SiU|S.U)-(flfA)(SiU)'} - V ((SAX/SxU m)'{8,S,){SMy]
+ \\. {(SiSi) (SMY - (SA) (SiUW = 0.
Let a/fi and a'/^S' be the two roots of this quadratic for Xi/Xj. Then
a8, + ^8^ and a'S. + fi'S^
are two reciprocal systems with their central axes at right angles, and there
are only two such systems belonging to the group SiS^.
(2) It follows from § 330 (3) that the central axes of these two systems
must intersect. Hence they intersect at a point on the axis of the group.
Let this point be called the ' Centre of the Group.'
Also let the two central axes of these systems be called the ' Secondary
Axes of the Group ' ; and let the systems Si and 8^ be called the ' Principal
Systems 'of the group.
Let the plane through the centre perpendicular to the axis be called the
' Diametral Plane of the Group.'
383. The Ctlindroid. (1) Let the assemblage of central axes of a
dual group, each axis being conceived as associated with its pitch, be termed
a CylindroiA (C£ Sir R. S. Ball's Theory of Screws.)
Take the centre of the group as the origin e, let eui and eu, be the two
secondary axes of the group and eu^ the axis of the group. Then ^i
eUf, eu, are a system of rectangular axes. Assume that i^, t^» t^i are unit
vectors.
Let cr, and tj, be the pitches associated respectively with eui and eu,, then
we may write
Si — eui + wigwi =eui + fffiU^v^,
(2) Then any other system S of the group, with its principal vector of
unit length and making sin angle 0 with iii, is /Sficos d + Sf^sin 0,
Hence
/S = e(wiCosd + w,8ind) + BriCosd. u,u, + tsr, sin ^ . u,u,
= (6 + fst«») (wi cos d + ti, sin tf)
+ («ri cos tf + f a sin 0) «^w, + («r, sin 5 — f , cos 0) u^u^ ;
where {, is a quantity which can be expressed in terms of tvi, «r,, and 0.
Now assume that the central axis of S cuts eui in the point e + f t^, and
that iff is the pitch of S.
Then (wicos ^4- ft sin tf ) t^a^s + (^r, sin tf — f g cos tf) i*,t*i
= «^8 (^ cos tf + 1£ J sin 5) = w cos tf . u^u^ + tsr sin ^ . v^u^ .
Hence 'CJi cos d + ft sin ^ = «r cos 0,
w, sin tf — ft cos tf = «r sin 0,
\
333—335] THE CYLIKDHOID. 533
Thus flr=:«riC08'd+«rg8m*d,
and ^, B (wi — tJi) sin 9 cos d.
These equations completely define the cylindroid.
(3) If ^ = e + ^i^^i + ^8<^ + ^lUz be any point on one of the central axes,
the equation of the surface, on which it lies, is obviously
(4) The director forces of the group are the two systems of zero pitch.
Hence the angles their lines make with etii are given by
Wi cos' ^ + Wa sin* 5=s 0.
It follows that the directrices of the group are real if ^ivr^ be negative,
and are imaginary if tJitj, be positive*
The distances from the centre of the points of intersection of the
dii'ectrices with the axis are easily seen to be ± V("" vritsr^),
334. The Harmonic Invariants. (1) The harmonic invariant H(x)
[cf. § 179 (3)] of any point x can be written 2xSi . S^ ; where Si and /S, are the
principal systems of the group.
Hence, with the notation of § 333 (1), by an easy reduction
H{x)^2 (xeuitii) e + 2m ^ {xeu^u^ Ui — 2«ri {aeit^Ut) u^ + 2flriflra (aniiUath) w,.
(2) Thus if X lie on the plane euiiUi, H{x) is a vector.
If a; be the centre of the group, the harmonic invariant is the vector u^
parallel to the axis of the group.
Also the harmonic invariant of ii, is the centre of the group.
(3) Also by a similar proof the harmonic plane-invariant of euiti, is the
plane at infinity ; and therefore conversely the harmonic plane-invariant of
the plane at infinity is the diametral plane of the group.
Hence H (U) = Xetinh-
This can be easily verified by direct transformation.
For H (U) = 2U/8fi . jSj = 2ui (ei*, + m^u^Ui) = - 2eu{u.2.
335. Triple Groups. (1) Any triple group defines a quadric surfistce,
called the director quadric. It was proved in § 186 (4) that if p, pi, p^, p^ be
the vertices of a self-conjugate tetrahedron of this quadric, then a set of
three reciprocal systems are given by
Si^ppi + ihP%Pz,\
Si^pp2 + fhPtPi\ (i),
St^pPt + fhPiPt,)
where fii, fhy fh ^^ three numbers, real or imaginary, depending on the
given group.
534 VEcnoRS. [chap, il
Also oDe of the vertices of the tetrahedron may be arbitrarily chosen.
Let the point p be taken to be the centre of the quadric, then pi, p^, p^
are on the plane at infinity, that is to say, they are vectors. Hence if
the centre of the quadric be called c, and Vi, Vg, Vj be unit vectors in the
directions of three conjugate diameters, a set of reciprocal systems can be
written,
S2 = cVi + fjL2VsVi,> (ii).
Thus if Si, 82, /S, be any three reciprocal systems of the group, SiVL, S^IX,
/Sf,U are the directions of a set of three conjugate diameters of the quadria
(2) If Ui, U2, U2 be three unit vectors in the directions of the principal
axes of the quadric, then the three corresponding systems,
will be called the principal systems of the group.
336. The Pole and Polar Invariants. (1) The polar invariant of
the point x is denoted in § 183 (3) by P (x) and is the polar plane of x with
respect to the quadric [cf. § 185 (1)]. Similarly the pole invariant of the plane
X is P (X) and is the pole of the plane X Thus the centre of the quadric
is the point P (U).
(2) Now let iSfi, fifj, St denote a set of reciprocal systems.
Then P(U) = 2U/Si.flf,.S, = 2/giU.S,.S,..
But 2iSiU . ^2 is the diametral plane of the dual group SiS^ [cf. 334 (3)].
Hence the centre is the null point of this plane with respect to jSf,.
Accordingly the diametral plane of any dual subgroup contained in the
triple group passes through the centre of the quadric. Also the centre of
the quadric is the null point of such a diametral plane with respect to the
system of the triple group reciprocal to the corresponding dual subgroup.
(3) It may be noticed that, if Hi (x\ H^ (x), H^ (x) denote the harmonic
invariants of x with respect to the dual groups SJS^, /S^Sfi, SiS^ respectively,
then [cf. § 184 (2)] \c = P (U) =: J?, (S^Vi) = H, {S,U) = J?, (iS,U).
These are four alternative methods of expressing the centre of the
quadric.
(4) Since the diametral planes of the three dual groups are
c./S,U./Sf,U, c.iS,U.SiU, c.s;u.iSf,u,
it follows that they intersect in pairs in the edges c . SiVi, c . S^Vi, c . SjiX.
i
336 — 338] THE POLE AND POLAR INVARIANTS. 535
(5) The diametral plane, which bisects chords parallel to any vector v, is
obviously P {v). Thus the diametral plane, of which the conjugate chords
are parallel to any line Z, is P (LU).
The diametral plane parallel to any vector area M is P (U) M; the vector
parallel to its conjugate chords is P {P (U) M].
337. Equation of the Associated Quadric. (1) The condition, that
any point p is on the quadric, can by § 187 (6) be written in the form
P(p)p = 0.
Now let pi be the length of the semi-diameter c . SiVU Then the point
p = c + piVi is on the quadric.
Hence by § 335, equations (ii),
And pSi.S^.S^.p-- (cViV^VtY {fhf^^ + /*i/>i'}'
Hence n«- u«u. - 1 i8.8,){8,8,)(SMy ^
Hence p^ ^ -^^. ._ ^_^_^_^^^,
with similar expressions for p^ and p,.
(2) Let Si, 2s, Ss be the principal systems of the group, and Ui, u,, <£,
unit vectors parallel to the principal vectors of Si, ^, Ss.
Let it be assumed that .
Si = ct^i + «r,gwi = 0^1 + ^Titijii,,
Sj = CU» + «ragti, = Ct^ + WaMji^ ,
S, = CU, + flTj^M, = CM, + flTjttiUa.
Let 0? be the point c + ^Hh + ^^^ + fs^ ; then fi, {„ {, are the rectangular
co-ordinates of a; referred to the three axes ct^, cus, ciis.
Also P(a?) = 2a?Si.S,.S,
= — 2 {wifiCUjM, + vJ^^SU^Ui + flTtffCUit^ — «ri«ra«rgtiitl,ti,}
= 2 I {«rif it*i + t!r,f jMa -I- vrJ^zU^ + WitSTj^sc} ;
where c is the centre of the rectangular normal system.
The equation of the quadric is {xP{x)] = 0, that is
fSr^GT^ ISTgCT] CTiCTj
338. Normals. (1) The vector parallel to the normal at a point x on
the quadric is 8 (-P W Uj ; and this can be transformed into \P {x — c), where
« is a unit point, and c is the centre of the quadria
The first expression requires no proof, if it be remembered that P (a?) is
a planar element in the tangent plane at x.
536 VECTORS. [chap. 11.
(2) To prove the second expression, we have
Now, c . P (a?) U = (cU) P (a?) - {cP (x)} U = P (a?) - {a? P (c)} U.
But P (c) = \U. Hence {xP (c)} = X.
Hence % {P(x) U} = |P(^) - X |U = |{P (x) -P(c)} = \P(x- c).
Thus the plane P{x — c) is the diametral plane perpendicular to the
normal.
339. Small Displacements of a Rigid Body. (1) If any rigid body be
successively displaced according to the specifications of two small congruent
transformations [cf. § 268 (1) and § 303], it is obviously immaterial which
of the two transformations is applied first, so long as small quantities of the
second order are neglected.
Now let e be any origin and eu^, eu^y eu^ any set of rectangular axes,
Uu u^, Ui being of unit length. Assume the three small translations
defined by the vectors Xocit^, Xa,u,, XOfUs, and the three small rotations with
axes eui, ev^y eu^ through angles XSi, XSsi XS, successively applied in any
order; where X is a small fraction whose square may be neglected, and
«!, a„ a„ S,, 8,, S3 are not necessarily small.
(2) Then any point x=^e + Sfw becomes Kx, where
ifa; = e + (f 1 + Xttx + XSaf a - XSjf «) i^ + (f 2 + Xo, + \B^i - XSif ,) u^
+ (f, + Xo, + XSif, - XSafOti,.
Hence the combined effect is equivalent to the combination of the small
translation X (oti^i + Oai/, + ci»^), and of the small rotation round an axis
e {BiV^ + Sji^a + S,i^) through an angle X V(8i* + 82' + ^j')* But by properly
choosing a^, a,, a, and Si, S^, Sg these can be made to be any small translation
and any small rotation with its axis through e.
Accordingly the above linear transformation is equivalent to the most
general form of small congruent transformation.
(3) Let S denote the system of linear elements
ttil^M, + OiViUri + Ostitis + SlCVq + S^filli + Sj^,.
Then x8 = (a, + Sjf , - 8,f ,) ev^th + (02 + Sjfi - Sif ,) ei^Ui
Hence Za? = a; + Xg (/Sa? . U).
Let S be termed the system of linear elements associated with the trans-
formation, or more shortly the associated system. And conversely XS will
be used as the name of the transformation.
(4) If two small congruent transformations \Si and XgiS, be successively
applied, then neglecting small quantities of the second order the combined
effect is that of the single transformation XiS^ + X^^,
^
339, 340] SMAI.L DISPLACEMENTS OF A RIGID BODY. 587
For Kix^sc + Xig {S^x . U),
neglecting W.
It is now obvious that evevy theorem respecting systems of linear
elements possesses its analogue respecting small congruent transformations.
(5) When 8 is & single linear element through any point e, the trans-
formation \iS is a rotation with its axis through e.
When £f is a vector area, the transformation AjS is a translation in a
direction perpendicular to the vector area.
The transformation XiS can be decomposed into two rotations roimd any
two conjugate lines of 8.
(6) Let 8 be written in the form av + «rSv, where v is a unit vector, then
Kx^x-i- \%{avx . U) + \«rS [%v.x. U}.
Now %[%v.x.Vi]^%{%v] = v.
Hence Kx ^x + \% {avx . U) + \wv.
Now Xav denotes a rotation through an angle \ round the axis av; and
\v%v denotes a translation parallel to v through a distance Xcr.
Thus the axis of the transformation is the central axis of the associated
force system, and the pitch of the force system is the ratio of the distance
of the translation to the angle of the rotation. Let this pitch also be called
the pitch of the transformation.
(7) Whatever point x may be, we may write 8 in the fonn XV'\-M.
Hence Kx = a? + \%M, In other words, \Sx . U defines the translation re-
quired to bring x into its final position.
340. Work. (1) If the force F pass through the point x, and x be
displaced to ^ -f- Xv, where v is a vector and \ is small, then the work done
by i?" is X (F^v),
This is obviously in accordance with the common definition of work.
The work can also be written in the equivalent forms, X(^|i;) and
X(t;|jPU).
(2) Let F=xu, and let \v be the displacement of x produced by the
congruent transformation \8.
Then t; = g(iSa?.U).
Hence %v = 8x,Vi.
And the work done by jP is
X {arw . (iSa? . U)} = - X {w . a? (/Sfa? . U)} = - X {m . x8} = X (F8).
It then follows, that the work done by a force F during the small con-
gruent transformation XS is the same at whatever point of its line of action
jP be supposed to be applied.
538 VECTORS. [chap. II.
(3) Let Fij F3, etc. be any number of forces acting on the rigid body
during its transformation. Then the sum of the work done by them is
where S' is the force system F1 + F2+
Hence the work done by the force system 8' during the small congruent
transformation \8 is equal to that done by the force system 8 during the
small congruent transformation T^.
Also if 8 and S be reciprocal, the work done is in both cases zero.
CHAPTER III.
Curves and Surfaces.
341. Curves. (1) Let the point a? (= « + 2f 2^) be conceived to be in
motion, so that fi, fg* fs ^^ continuous functions of the single variable r,
which is the time.
Then x + xZr = 6 + S^ + hrX^u, is the position of the point at the time
T + St. Hence the vector A, which is S|i^, represents the velocity of x in
magnitude and direction.
Similarly the vector x, or S^u, represents the acceleration of a; in magni-
tude and direction. Let x, x\ etc. be formed according to the same law.
(2) Let a be the length of the arc traversed during the time r, and
cr + Scr that traversed during the time t + 8t.
Then ^ = ^/(xy.
(3) The tangent line to the path of x at the time r is ^, that at the
time T + 8t is aac + xxSt,
Let Se be the angle of contingence between these tangent lines, then by
§ 322 (2)
V ii>y
— w~^-
(4) Hence, if /> be the principal radius of curvature of the path,
1 _ (ie _ is/(xx \a£)
p-d^- {(*)•}» •
Therefore
1 (ix lis) _ (B IB) (A {ay
540 CURVES AND SURFACES. [CHAP. 111.
342. Osculating Plane and Normals. (1) At the end of a second
interval Br, x has moved to a? + 2xSr + 2x (St)*.
Therefore the osculating plane is ocxx.
The vector factor of this product is xx. Hence the direction of the
binormal is that of the vector %xx. The binormal is x^xx.
(2) The neighbouring osculating plane is x(xx+xx8t). The vector
factor of the neighbouring binomial is g (xx + xxBr). Let SK be the angle of
toi-sion.
Then [cf. 322 (2)] ^^^^^3^^^,^).
^ince %xx . g {xx + xxBr) = %xx , %xxBt.
Now by § 326 (4) equation (iv), %xx . gi?ic = (exxx) ^x.
Hence — = /(?^^f^L^ = (exxx) ^{xy
dr V (xxy (xxy
Now let — be the measure of the torsion.
K
K {xx \XX)
Thus — = ^— *^
(8) The normal plane at x is x%x.
The principal normal at a? is the line N, where
N = XXX . X^X = {XX%X) XX — {XX%X) XX = {xy XX ^{x\x) XX,
The vector parallel to the principal normal is therefore
(xyx — (x \x)x.
343. Acceleration. In order to resolve the acceleration along the
tangent and the principal normal, notice that
_ x\x) . {xyx — {x\x)x
Now let t and n be unit vectors along the tangent and principal normal
respectively.
Then x = {d^yt, (i?)«^-(i?|^)A = ^'n.
Hence x = h-r^r t + ^^^n.
{{xy}^ p
Therefore the acceleration is equivalent to a component j along the
(xY
tangent, and ^-^ along the normal.
r
342 — 345] SIMPLIFIED PORMULiE. 541
344. Simplified Formula. (1) In order to develope more fully the
theory of curves^ let us make use of the simplification introduced by the
supposition that the curve is traversed with uniform velocity by the moving
point. We may then take cr as the independent variable, and use dashes to
denote differential coefficients with respect to cr.
(2) Collecting our formulae, we have in this case
(aiy = l, and therefore (a/\a/')=^0, (a/y+(a/\a/'')^0.
The tangent line is auc' ; the normal plane is x%a!' ; the osculating plane
is xafa/' ; the binormal is a?gfl?V' ; the principal normal is xsc".
(3) To find Monge's polar line of the point x of the curve, notice that
the normal plane at the point x+a/Sa is (x + a/Sa){^af + %a/' .B(r},thB,t is,
x%a/ + (x^x'' + U) Scr, since (a/%af) = U-
Hence the polar line is x%x' . (x%x'' + U), that is, a%x . a%a/' + %x\
It is obvious that x%x' ,a%x" is some line through x. Assume that it
is ocv.
Then i; = a:t;. U =a?8a?' .arga?". U = a;g^' .ga:"= ga?'a?",
by § 326 (5). Hence the polar line of a? is x^afaf' + ^x,
(4) The centre of curvature is the point where this line meets the
osculating plane, that is the point
{x^afaf' + ^x') . XX od' = (a? V')» x\%x' . xafx'\
Now '^af.xafx''=^'-'\x\\k.xafaf'=^\x\afaf'^'-{a/'\x')af'\-{x\af)af'^
x"
Hence the centre of principal curvature is the point ^+7^7v'^» ^^^^ ^®
the point x + f^a/\
345. Spherical Curvature. (1) The centre of spherical curvature
is the point where three neighbouring normal planes intersect, that is the
point
by use of the transformation of § 344 (3).
Now by the rule of the middle factor [cf. § 102 (7)]
a%afx" . a%x'" = {a%x'af' . ^af") x = {a%a/''%a/a/') x.
Also g«'"ga?'a?" = I (ea/'' . exW') = (earV V'O | e = (eafaf'af'') U.
Hence a%a/a/' . a%a/" = (ea/a/W) x.
Also by § 326 (5) %a/ .a%a/'' --^a/'W ^--^a/af".
Hence the centre of spherical curvature is
542 CURVES AND SURFACES. [CHAP. III.
(2) Therefore, if /9i be the radius of spherical curvature,
p,» = pV {x'af'J = pV [{afy - {of \x"y]
(3) Now {af'y can be found by squaring the determinant {eafaf'x'").
Thus [cf. § 342 (2)] -^ = (ea?'^V'' \eafx"a!")
1. 0. («i lo
0, {a!'\a!'\ («"|0
Now i = (a,"|«") = -(«'k"). And -|g = 2(<r"|<r"'X
Hence 1 _ jafj 1 /dpV 1
p^H^ p* p^\d<r) p**
Therefore Pi' = P' + '^ (^y + '^-"^ = P' + *' (^T-
This is the well-known formula for p^.
346. Locus OP Centre op Curvature. (1) It is easy to see that
the inner products of the various differential coefficients of x, such as af^ x'\
etc., can be expressed in terms of p and k and their differential coefficients
with respect to cr.
Let px be written for a /]p' + '^^ \'j- ) \ •
Then we have by successive differentiation
(«?')»= 1, (fl?>'0=0» (O' + (^'lO=0,
3 {x" \x'") + (a;' |a^0 = 0. 3 (^"? + 4 (a?" \ai') + (a?' |arO = 0,
and so on.
Also (xy^\^^(a!\(x!'\ Hence (a?" |0 = - A ^ = -^(^''^0.
Again («.'T = ^ + ^i = X«, say. And (^"VO = xg.
Also ^(-'10 = (-''')--f(-"K) = -^(^.|).
Thus (re" ja:*^) is expressed in the required form, and so on,
346, 347] LOCUS of centre of curvature. 543
(2) Let y be the centre of principal curvature of the curve at the
point X Let a' be the arc of the locus of y measured from some fixed point.
Then when <r is the independent variable, and y is used for ^ ,
Hence (5~y-(y)' = l + V(^)\^? + V(^'lO
= 1+4
©■-«-H^y-s'-'-i'-
Therefore 5^ = ^ .
(3) Also if he be the angle of contingence corresponding to ha', then
d«' ^/(yy yy)
Thus by mere multiplication -,- , and hence ^, , is expressed in terms
of p and K and of their differential coefficients with respect to <r.
347. Gauss' Curvilinear Co-ordinates. (1) Let x be any unit
point on a given surface. Then the co-ordinates of x referred to any four
reference elements may be conceived as definite functions of two independent
variables 0 and ^. Then the two equations, 0 = constant, and ^ = constant,
represent two families of curves traced on the surface.
(2) Suppose that the unit point x + Sx corresponds to the values 0 + S0
and ^ + 5^ of the variables. Then Bx is the vector representing the line
joining the point (0, ^) with the point (0 + Stf, ^ + S^) of the surface.
Also Bx can be written in the form
Bx = (x^B0 + xJ5<l>) + i (xnB0\^ + 2xy^B0B<f> + x^B^\^) + . . . ,
where Xi, x^, x^i, ^g, x^^ are vectors. Hence, if e be the origin,
(e k) = 0 = (e |«0 = (e \xn) = (e M = {e \x^.
(3) In order to exhibit the meanings of these vectors, let e be any origin
and et^i, eu^, eut rectangular unit axe&
Then x is e-hSfw, and a?, is 2r|w, a?, is 2^w, Xu iaX^u, and so oij.
644 CURVES AND SURFACES. [CHAP. III.
It will at times be an obviously convenient notation to write ^ for iCj.
^^ for a?„ ^ for a?n, and so on.
(4) The distance ha between a? and x + hx\& given by
(Scry = (Sa;)« = (a?i |a?i) (S«)« + 2 (a?i 1^,) S«S<^ + (a:, |fl:0 (&/»>^^
(6) The tangent line of the curve joining the points is
X (xiS0 + x^Bif)).
Now let X + BiX be a neighbouring point on the curve ^ = constant, and
flj + Safl? on the curve 0 = constant.
Then x + BiX = x + XiS0+^Xn(B0y, x -{■ BgX ^ x + XiB(f> -\- ^ XtiB(l>\\
Hence two tangent lines to the curves ^ and 0 respectively are xxi and
xx^. Accordingly the tangent plane at x is xx^x^.
The normal at ^ is a%XiOs^.
(6) The angle to between the tangents at x to the 0 curve and the
<f> curve is given by
coQjj- (^^1^) sin©= / (^^ki^)
(7) Let Bv be the perpendicular from a; + &i? on to the tangent plane.
en[c.§ ( )] V(^,a?, \x,x;) " VK^ l^i) (^ li) - (a?i laJ*)"}
^ 1 (eg-taycii) (Sgy + 2 (ea;iavg,a) B0B<f> + (ex^x^^d (^Y
2 \/(^1^2 ,^l^s)
348. Curvature. (1) Let p be the radius of curvature of the normal
1 (BcY
section through x and x + Bx. Then p = s -^ .
H^^^^ V(aH^ kia?g) . {{x, \x,) {B0y + 2 (x, \x,) B0B(f> + (a;,|a?,)(S»y}
'^ (exiX^i^ (B0y + 2 (exiX^fiPja) B0B(l> + (exix^^ (S^)*
(2) Now seek for the maximum and minimum values of p when the
ratio of B0 to B<f> is varied. Let pi and p^ be the maximum and minimum
values found, and let B0i to B^ and BO^ to S^ be the corresponding ratios of
B0 to B^, Then pi/V(^i^l^^) and p2l\/{xiX^ x^x^ are the roots of the
following quadratic for 2^:
Hence — = (^i^'V^O i^^^^td - (^^^"^^itY
348, 349]
CURVATURE.
545
(3) The expression for — can be put in terms of (x^ \a^\ (x^ la^), (x^ |ajj),
and of their differential coefficients with respect to 0 and ^
For, since by § 847 (2) (e k) = 0 = . . . = (e |«b),
(ex^x^^)(exix^^(exix^n\exix^=:^ (aa|«i), (a^\x^), (ah|^)
(^|«i), (a^ilaJa), (os^lxn)
(a^nk), («ii|^), («?n|fl!«)
(a?i|a?i),. (^iki), (aJilar^)
(a?i|^), (^sjaJa), («j|«ia)
(a?i|a?i2), (a?,|a?i2), (aqja^n)
Hence (ex^x^^) (exiO^o) — {exiX^^^Y
and
(ex^x^^f =
(^i|«i). (a?i|fl:.), (a?i|a^)
(^|a^), (a^ka), (^iaJjfi)
(^ki), (^l^ji), 0
(aik), («i|aJ,), («ija?is)
(^il^a), (^al^Ji), (^|a?ia)
+ {(^ ki) (^ 1^) - (^ 1^)"} {(^11 1'-^aa) - (^a |^«)}.
Now let (x^ |fl^)i, (a?j \xi)^, etc. stand for ^^ (xi \xi),
d (xi |a?,)
, etc.
Then (x^ la^X = 2 (ah la^i), (a?i l^^ih = 2 (a^ |a?ia),
(a?s |fl?jX = 2 (asa |a?i«), (a?2 1«2)» = 2 (a^ |a^),
(a?, |a!,), = (ahi |a?j) + (a?, |a:ia), (a^ \x^\ = (aru ka) + (^ |«?«),
(a?! |a?i)jH = 2 (a?u la?,.,) + 2 (a?i |a?iaa), (^tj |a^)u = 2 (a^a |a?,a) + 2 (aj^ |a?„j),
(a?i la^aXa = (^u 1^) + (^a kia) + (^la | ^) + (^i |^iaa)«
Hence (a?u la^ag) - (a?u |i»ia) = (^i ka)u - i («a k)a - i (a?a laJaXi-
Thus — can be expressed in the required manner.
P1P2
349. Lines of Curvature. (1) Also if fi and f, stand for
/>i/V(^i^j|^i^) a^d /5a/V(^^2 l^i^s) respectively.
Then the ratios 8^1 to S<^ and 80, to 8^ are given by
(6X1X^2) ?i - (^1 l^j) "" (ed?,aviai) ?i - (aq |a?,) '
and ^ = -'Ah .
(ex^x^;) fa - (a?! |aji) (exiX^^) & - (a^i |a?i) '
(2) By the aid of the quadratic for f, and J; which have been found,
it can easily be seen that the lines of curvature are given by
{{Xi [x^) {ex^x^u) - (^1 i^) («a?iavi?ia)} (8^)'
+ {(a^a I ^a) (ca?iavcii) - (a?i \x^ ifix^x^c^)] B0B<l>
+ {(asa ka) (exix^^) - (a?, 1 aja) (ex^x^} (8<^)" = 0.
W. 35
546 CURVES AND SURFACES. [CHAP. IIL
And therefore it follows that
{X, \a^) S0iB0^ + {x^ la;,) [B0^S<f>2 + S^aS^^} + (^a k) S^S<^ = 0.
(3) Let x+Sx and x+ S'x be two neighbouring points to x, where
&i? = a;i8d + axaS^ + .'.., and B'x=^XiB'0 + x^'4>+ ^^ •
Then the angle yft between the two tangent lines x8x and xS'x is given by
(Sx\b'x)
^^^'^^^[(Bx\8x){S^w\S'x)}
_ (x, ifl?,) h0^0 + (x,\x^) (S0S'<t> + Sil>B'0) + (x^ \x^)Sil>y<l>
. . {S0S'<l> - B'0Sd>) ^(xix^ \x,x^)
sini;r=^ ^ y ■ — .
oa-oa-
Hence
ScrSV cos -^fr = (x^ \x,) B0S0 + {ofi l^a) (Stf S'^ + B'0S<l>) + (as. jasg) Si/^y^,
and [cf. § 347 (6)]
So-SV sin i|r = (S^S'i^ - &'0B4>) sin a> V{(^ k) (^ l^a)}.
Corollary, The tangents to the lines of curvature at x are at right angles.
(4) The conditions that the 0 curves and the <f> curves should be lines
of curvature at each point are, from subsection (2) and the corollary of
subsection (8), that the equations,
{oh M = 0 = (exix^i),
should obtain at each point.
360. Dupin's Theorem. (1) Let x be conceived as a function of
three variables 0, ^, y^. Then the equations 0=^ constant, ^ = constant,
and yjr = constant, determine three families of surfaces. On the surface,
0= constant, a; is a function of the variables ^ and ^; on the surfiu^,
<l> = constant, a function of yff and 0 ; on the surface, y^ = constant, of 0 aud ^.
^^ S^~^'* ^~^' ^~^" ^^^^ * corresponding notation for the
higher diffei'ential coefficients.
(2) Now suppose that the three families of surfaces intersect orthogonally.
Then (a?i|««) = 0, (x^\x,)--0, {x^\x,)^0 (1).
Hence by differentiation
(a?i8 |a^) + (a?i laJas) = 0, (a:,a la?,) + (ar, |abs) = 0, (x^ la?i) + (a^ |a?ia) = 0
Hence (x^ \^)-0 = (x^ \^n)=(^s l^w) (2)-
The condition that the lines of intersection of 0 and ^ with the sur&ce 'f
should be lines of curvature is {ex^x^^ = 0.
But from equations (1) x^x^ = ^ar,, and hence we may write x^x^ = XR^Tj.
And therefore from equations (2)
(exix^ai^) = X (egaJj. Xi^) =* X (a?, | x^) « 0.
350 — 352] dupin's theorem. 547
Hence the lines of intersection are lines of curvature of the surfaces on
which they lie.
351. Euler's Theorem. (I) Let the curves 0 == constant and
^ ss constant be lines of curvature, so that
Let pi be the radius of curvature of the normal section through «d?, and /a,
of that through osx^.
The radius of curvature of any normal section is given by
P P\ fh
(2) The angle '^, which the tangent line xix makes with the tangent
line xxi, is given by
Hence 1 ^cos-^^^nl^^
p h pi
352. Meunier's Theorem. (1) The principal radius of curvature of
the curve, ^ = constant, is
,, ^^ V — ^ = iP> say-
The radius of curvature of the normal section through xx^ is
^ — (^^;s^"-
Hence ^ = (??^^»)K^)'L* .
(2) The osculating plane of the curve ^ is xx^po^, the normal section
is xXi%XiX2. If X be the angle between these planes, it is given by
But a?iSa?,a?a = (a?i | asg) 8«i - i^i I «i) 8^^-
Therefore (a?i8^a?s | aJia?ji) = (a?i | a?i) {ex^x^i).
And (a?i8^ajj)" = C^)* (^1^:2 1^^).
XT {(^i)*}* (eXjPi^i) ip
Hence cos v = ^,,/ ^ ^. . v^, = - .
^ y/[{x^x^y{XiX,,y} p
Note. I do not think that any of the formulae or proofs of the present chapter
have been given before in terms of the Calculus of Extension.
35—2
CHAPTER IV.
Pure Vector Formula.
353. Introductory. (1) A simple and useful form of the Calculus of
Extension for application to physical problems is arrived at by dropping
altogether the representation of the point as the primary element, and only
retaining vectors. The relations between vectors of unit length will give
the expressions in terms of the Calculus of Extension for the formube of
Spherical Trigonometry. Also many formulsB of Mathematical Physics can be
immediately translated into this notation. These vectors may [cf. § 210 (3)]
also be conceived as the elements of a two-dimensional region^ and their
metrical relations are those of two-dimensional Elliptic Geometry.
(2) Let iyj, k represent any three unit vectors at right angles. Then
any other vector x takes the form fi + 1][; + ^k.
(3) We will recapitulate the forms which the formulse assume in this
case. It will be obvious that, as stated above, they form a special case of
Elliptic Geometry.
|jA? = i = tl*» l**=j = lii^ \ij^k = \\k.
U\fc)^0 = (k\i)^ii\j).
{i\i)^U\j) = (k\k)^(ijk)^h
(4) The multiplication formulse are the ordinary formulae for a two-
dimensional region : we mention them all for the sake of convenience of
reference.
Let X, y be any two vectors ; and X, Y any two vector areas.
Then xy^ — yx, XF = — YX.
Also xy represents a vector area, and XY a, vector: the vector area ay
is parallel to both vectors x and y, and the vector ZF is parallel to the
intersection of the vector areas X and F,
353 — 355] INTRODUCTORY. 549
Again. («|y)-(yl«). {X \T) = (Y\X);
and the result is in each case a purely numerical quantity.
364. Lengths and Areas. (I) The length of the vector x is ^(x |^).
The angle d between two vectors x and x' is given by
^^ ^ " ^[(x \x) (x' \afj} ' S"^ ^ - y {(a. \x) (of \a/)} '
(2) Any vector area X takes the form ^k + rjki +
The magnitude of the area is */(X [ Jf).
The angle 0 between two vector areas X and X' is given by
cos
e^ - (^-i^l_- sin5= /_ixr]xx2_
^[{x\x)(X'\x')}' """ \/{(x\x)(X'\x')y
(3) Also \X denotes a vector line of length ts/(X |X), and \x denotes a
vector area of magnitude V(^ k)*
It will be useful at times to employ the term ' flux ' to denote a vector
area.
(4) Let { and 17 be the lengths of the vectors x and y, and let 0 be the
angle between them. Then the magnitude of the vector area ony is
V{«y l«y} = V{(a? k) (y |y) - (^ lyY] = ^7 sin ^.
Again, let { and 17 be the magnitudes of the vector areas X and F, and
0 the angle between their planes. Then the length of the vector XF is
^\XY\X7]^^{(X\X)(Y{7)^(X\Yy]^hBm0.
356. Formulas. (1) The extended rule of the middle fiactor [c£ § 103]
gives the following formulse :
X .xy^(Xy)x--(Xx)y (i);
x.XY^(xY)X^{xX)Y (ii).
The second can also be deduced from the first by taking supplements.
(2) The same rule also gives the following formulae for inner multipli-
cation :
xy\z=^{x\z)y''(y\z)x (iii);
z\xy^z,\x\y==(z\y)\X'-{z\x)ly (iv).
(3) Also from §105,
xy.XY^(xX)(yY)-(xY)(yX) (v).
And writing \u for X and jt; for Y, we deduce
(xy\uv)^(x\u)(y\v)'-(x\v){y\u) (vi).
Similarly, the supplemental formula
{XY\UV)^(X\U)(Y\r)^(X\V)(Y\U) (vii).
550 1»URE VECTOR FORMULJi. [CHAP. IV.
(4) Two particular casf^ of the formulsd (vi) and (vii) have been already
used above, namely,
{ayy\scy):=(x\x)iy\y)-(x\yy (viii);
(ZF|Z7) = (Z|Z)(7|F)-(Z|F)« (ix>
It will be convenient to write any expression of the form (jc\x) in the
form {xf or a^, and {X\X) in the form (X)» or X\ Thus {xyY stands for
(ay |ajy).
356. The Origin. By conceiving, the vectors drawn from any arbitrary
origin 0, any vector x may be considered as representing a point. Thus it
is the point P such that the line from 0 to P can be taken to represent the
vector X in magnitude and direction. This origin however is not symbolized
in the present application of the Calculus.
367. New Convention. (1) Before proceeding with the development of
this Calculus it will be advisable explicitly to abandon, for this chapter only,
the convention [cf. § 61 (1)] which has hitherto been rigorously adhered to,
that letters of the Italic alphabet represent algebraic extraordinaries and
letters of the Greek alphabet numerical quantities of ordinary algebra.
As a matter of practical use and not merely of theoretical capabilities it
would be found so necessary by any investigator in mathematical physics to
continually form the Cartesian equivalents of his equations — if only for
comparison with other investigations — that the capabilities of the Greek
alphabet for the representation of nuignerical quantities would not be found
sufficient. The German alphabet is found by most people difficult to write
and to read. But let the following convention, which is a modified form of
one adopted by Oliver* Heaviside, be adopted.
(2) Let all letters of the Greek alphabet denote numerical quantities.
Let all letters, capital and small, of the Latin alphabet tvithout svhscripts
denote respectively vector areas and vectors; except that in formulae con-
cerned with Kinematics or with Mathematical Physics t always denotes the
time.
Let i,j, k denote invariably three rectangular unit vectors.
If X denote any vector, let Xi, x^fX^he numerical qiiuntities denoting the
magnitudes of the resolved parts of a; in the directions %,j, k respectively:
so that x^Xii + x^j + xJe.
Let iTo be a numerical quantity denoting the magnitude of x. Thus
Xo = ^/(xy = V(«i» + a?,» + a-3»).
• Cf. * On Ihe Forces, Stresses, and Fluxes of Energy in the Electromagnetic Field,' Phil
Trans. 1S92.
356 — 359] NEW CONVENTION. 551
If X denote any iluz, let Xu X^, X^h^ numerical quantities denoting the
magnitudes of the resolved parts of X along the unit fluxes jk, ki, ij: so
that X = Xijk + XJci + X^j, Let Z© denote the magnitude of the flux, so
that
Zo = + sl{Xy = {Z,» + Z,» + X,^]\
(3) This notation avoids a too rapid consumption of the letters of the
alphabet, and shews ,at a glance the relationships of the various symbols
employed.
We note as obvious truths ; i
if Z= |ir', then Zo = a?o> Zi = iFi, X^^x^i, X^ — x^.
Also we note that if a?=|Z, then \j»=,|X = Z, and the same results
follow.
358. System of Forces. (1) Let forces represented by the vectors
/, /', ..., act at points denoted by the vectors x, x\ ..., drawn from any
assigned origin.
Then any force/ at a; is equivalent to/ at the origin and a vector area xf
representing the moment of/ at x about the origin.
(2) Hence the system is equivalent to S/* at origin and the vector area
^f, representing a couple ; which may be called the vector moment of the
system about the origin.
If L be this 'vector area,' L^ Z„ i, are the three moments of the
system about axes through the origin parallel to t, jy k.
369. Kinematics. (1) Let any point in space be determined [cf.
§ 350 (1)] by the three generalized co-ordinates {6, (f>, '^). It will be called
the point (6y ^, yft). If the point be referred to three rectangular axes, the
rectangular co-ordinates will be written Xi, x^, x^y and the point will be
represented by the vector x.
If 0, ^, -^ be conceived as the co-ordinates of a moving particle, they are
functions of the time. Let u be the vector which denotes the velocity of
the particle at each instant ; then corresponding to each position {6, <f>, yjr)
there is a definite velocity u. Hence u must be conceived as a function of
dy^fi^: that is to say, if t, j, k be any three fixed rectangular vectors, and
u^Uii + u^j + ujcy then Ui,u^,u^ are functions of 0, ^, '^,
Since 0, ^, ^ are functions of the time, u can also be conceived as a
function of the time.
(2) Let u be the velocity of the point at the time t, and u + vZt at the
time t + ht Then when Zt is made infinitely small, u is the acceleration.
552 PURE VECTOR F0RMULJ5. [CHAP. IV.
Also evidently,
4« --S 4#-<9L ^. 4iL. /l ^ 4/^2* S=
... . • , . f du
(3) The aspect of the osculating plane of the curve traced by the point
is represented by the vector-area uu.
The binormal is represented by the vector \uu.
The normal plane is represented by the vector-area | u.
The principal normal is represented by the unit vector n, where
uu\u _ (i^ |w) ti — (u \u) u
" »J{uu I m)" ~ tj[{u \u) {uu \uu)] '
The distance traversed in the short time St is *</(uy . St
(4) The angle Se between the directions of motion at the times t and
t + Stia given by
. -, /(uuluu) ^
V (uY
{uy
The radius of curvature is
{(")•}»
(5) Thus i = V^«l!^.„ + l!^„
_^(uy^^d^(uy u
p dt ' V(w)*
This represents the ordinary normal and tangential resolution of the
acceleration.
360. A Continuously Distributed Substance. (1) Many branches
of Mathematical Physics depend upon the investigation of the kinematical
properties of substances (ordinary matter or some other medium) distributed
continuously throughout all, or some portion of, space. The continuously
distributed substance will possess various properties dependent on its motion
and on other intrinsic properties. Let any quantity associated with a
particle of matter, which does not require a direction for its specification, be
termed scalar, according to Hamilton's nomenclature.
(2) Then scalar quantities, such as the density, and vector quantities,
such as the acceleration, are associated at each point with the existence of
the continuously distributed matter.
These quantities, scalar or vector, may be associated either with the
varying elements of matter occupying given points of space, or with the
given elements of matter occupying varying points in space.
(3) If the quantities be thus associated in the firat way with the given
points of space, then the co-ordinates — ^say tf, (f>, '^^— of any point are not to
360] A CONTINUOUSLY DISTRIBUTED SUBSTANCE. 553
be considered as functions of the time. Let x ^ ^^y scalar function of the
matter at the point {0, ^^ yft) at any time t, then at the subsequent instant
t + St Ek fresh element of matter occupies the position (d, <l>, yfr). Let its
corresponding scalar function at the time t-\-Bt be X"^^^^- ^^^^ X ^
conceived as expressed in the form x(^f ^> ^» 0> where 0, <f>, yjr ai*e not
functions of t The operators
d0' d<f>' dit' dt
applied to x ^^® therefore the relative properties of opeiutors denoting
partial differentiation.
Call ^ the stationary differential operator with respect to the time.
(4) Similarly if u be any vector function of the matter at the given point
{0, <f>, y^) at any time t, then at the subsequent instant t + St the correspond-
ing vector function of the new element which occupies the position (0, ^, yjr)
can be written w + -^r St. Also it is obvious that
dt dt . dt dt '
Let ^ and ^ be abbreviated into u* and v', or into i^, Ye-
ot ot /v /v
(5) We shall assume, except where the limitation is expressly stated,
that the scalar and vector functions spoken of are continuous functions of
the variables: and that if ;^ be any scalar and u any vector, X' ^' ^> ^
have finite and continuous partial and stationary differential coefficients with
respect to 0, <l>, yjr, t
(6) If the quantities be associated with the given particles of matter, let
the co-ordinates 0, ^, '^ mark the position of any given particle at the time t
Then at the subsequent time t + ht, the co-ordinates of that particle have
become 0-h dht^ <f>-\-^htj -^ +'^&. Also if % be any scalar fimction of that
particle at the time t, the same function of the same particle at the time
t-^-U will be denoted by ^ + XJ^t or by ^ + ^ S^- The function x ^^^ ^
conceived now as a function of 0, <f>, y^, t and written x (^» ^> ^> 0> where
0, ^, '^ are functions of the time. Thus
dt dt^d0 ^d<l>^^dy^^
Similarly if u be any vector associated with the particle, at the time t+ 8t
du
the same vector function associated with the same particle is u+uSt oru+'-^ St
554 PURE VECTOR FORMULiE. [CHAP. IT.
m, , . , du dut . . d'Uq . . du* ,
Thus obviously _=_.» + _^ + ^ A; ;
, dui dui ^ Sw, . 3te, ; dtii
with two similar equations. Hence
du du .du :du rdu^
dt dt^^d0^'''d4>^^d^'
Call T- the mobile dififerential operator with respect to the time.
361. Hamilton's Differential Operator. (1) Let the position of
any point be denoted by the vector w which is represented by the line drawn
to it from any arbitrarily assumed origin. Then ^i, o^, ^s are the rectanguJar
co-ordinates of the point referred to axes parallel to i, j, k] and a^j, 0S2, ^ niay
be conceived as taking the place of the unspecified co-ordinates ff, 0, V^ of the
previous investigations.
(2) Let X ^ any scalar function of position at a given instant. Then
i^ + j^ + i^ obviously represents the rate of change of x ^^ *^^ point x
in the direction of the normal to the surface ;^= constant, which passes
through a?. It follows that the function *o^^+i^ + *^ ^ independent of
the directions of the vectors i, j, k so long as they are a rectangular set.
O 'i o
Let the symbol operator, »V-+i^— 4-A;;r— , be written V, and called
da?i 0X.2 ox^
Hamilton's* Differential Operator, or more shortly, the Hamiltonian. Its
properties were first fully investigated by Prof. Tait** for the very similar
case of quaternions.
(3) The Hamiltonian may accordingly be conceived as operating on a
vector by means of the conventions
Vu = Vvri . i + V«j .j + Vti, . fc,
(V \u) = {Vu, 10 + (Vi^ \j) + (Vt^ \k).
I Vt^ ip called the Curl of the vector t^, Vu is the Curl-flux of the vector it-
(V |u) is called the Divergence of the vector u and is a scalar quantity.
*»
* Gf. Hamilton's Lectures on Quatemione, Leottize vii. §
Of. his Eleitientary Treatise on Quaternions, Ist Edition, 1867, 8rd Edition, 1890.
361, 362] Hamilton's differential operatoh. 555
(6) It is obvious ^vith this symbolic use of V that it can be treated as a
vector as far as formal algebraical transformations are concerned, so long as in
the product it is kept to the left of the quantity which it operates on, and so
long as those quantities to its right on which it does not operate are noted.
(7) Thus in accordance with the rest of our algebmcal notation we may
write V Vx in the form V«x» where [cf. § 367 (2)]
dxi* dx^ dx^
It is obvious that V^;^ = 0; VVm = 0. Again, V | V . t^ becomes V*u, which
is V*«^ . i 4- Vht^ ,j + V«a, . k.
(8) An important example of the possibility of formal algebraic
transformations of expressions involving V is as follows :
If a, 6, c be any vectors,
ofc jc = (a |c) 6 — (ft |c) a = — j(c |a6) = |(c jfta).
Hence (ft | c) a = ft (c ja) — | (c \ba).
Now putting V for both ft and c and u for a, we find
V«M = V(V|i£)-|(V|Vt«).
362. Conventions and Formula. (1) The symbol V is to be
assumed as operating on all the subsequent vectors in a product in which it
stands, in the absence of some special mark attached to a vector. If a
vector such as i; is not operated on by a preceding V, let it be written with a
bar on the top, thus v. For instance, Vuv implies that V operates both on
u and V ; but Vuv implies that V operates on u and not on v, and VUv implies
that V operates on v and not on u. Similarly V (u\v) implies that V operates
on V and not on u.
(2) The advantage of affixing a sign to a vector not operated on by V
is that, as far as the vanishing of a product is concerned owing to the formal
laws of multiplication [cf. § 93 (4)] a vector behaves diflferently according as
it is or is not operated on by V.
For instance if m, v, are any two vectors, uvu = vuu = 0. This is true by
reason of the formal laws of multiplication. Now substitute the symbolic
vector V for v, then uVu = Vuu, and this is not zero. Thus it is convenient,
as far as formal multiplication is concerned, to reckon u and u as different
vectors. It sometimes conduces to clearness in tracing the algebraic
transformations to preserve the bar over a vector even when it is placed in
front of V ; thus Vt^u = uVu. In such cases the bar may obviously be placed
or dropped without express mention.
556 PURE VECTOR FORMULiE. [CHAP. IV.
(3) The following are important examples :
V(u|t;) = V(u|t;) + V(tt|t;),) ...
hence V(i^|u) = iVw« J ^^
Vwt; = Vmv + Vuv = vVw — wVt; (ii).
Also [cf. 361 (8)]
it I Vw = - 1 (Vw . tl) = I V (u\u) - (u |V) |tt = ^ [Vu^-(u |V) I u.
But \(u \Vu) = [t* . Vw = - Vt^ |u.
Therefore Vm | u = (w | V) it - ^ Vw',| ^. . ..
or (w|V)M = Vit|tZ + iVu« j ^"^^
(4) If the operation V is repeated in a product, a little care must be
exercised so that the use of the bar may be unambiguous. For instance the
following transformation exemplifies this remark.
We wish to operate on Vu|u with V. The new operation of course
operates both on the u and the u of the existing expression ; since the bar
merely refers to the existing operation V. Write Vtt [tl == — |w , Vu.
Then V (\u. Vw) = V (|u . Vit) + V (,it . Vw),
where obviously the newly placed bars refer to the V outside the bracket.
(6) But the introduction of a new symbol, such as {v = Vt/ is often the
simplest solution of the difficulty. For instance Vu\u becomes \vu.
An important example is arrived at by operating with V| on Vu\u.
Write \v for Vu.
Then V | . [ vu = Vvu = uVv — vVu, by equation (ii).
But vVm = (t; I v) = v* = (Vu)",
and LiVv = u.V \Vu = (u j V) (V (tt) - w V*u.
Hence Vt;tt = (u|V)(V|u)-u|V*u-(Vu)* (iv),
where v — \Vu.
(6) It will be convenient to adhere to the further convention thatV
immediately preceding a scalar such as <f> does not operate on a subsequent
vector unless some stop is placed between the V and the scalar.
Thus V<f>u has the same meaning as V^ii, but V . <f>u implies that V
operates on <l>u. This convention is useful in dealing with such expressions
as V^V-^: it avoids the clumsy form V^V-^.
It is however often better to place bars where there is a risk of mistake,
so as not unduly to burden the memory with conventions.
(7) The preceding transformations have brought into prominence the
symbolic operator (u|V). It is a scalar operator, and in the CartesiaB
o o o J
notation [cf. § 357 (2)1 is Ui;^— 4-U2;r— +w,^- = Mo 3- > where da- is an
owi 0X2 oXi da
element of arc at the point x in the direction of u.
363] CONVENTIONS AND FORMULAE. 557
It follows that, if u be the velocity of the matter at the point x,
(U ot uXi 0W2 vnci at
363. Polar Co-ordinates. (1) The analytical transformations of V
into polar and cylindrical co-ordinates can be easily established.
Let P be the point a?, and 0 the origin : let the position of P be defined
Fio. 1.
by the length p of OP, the angle 0 between OP and the direction of Jfc, the
angle <f> between the plane through OP and k and the plane ki.
It may be noted that by the convention of § 357 (2) p has also been
denoted by a?o.
(2) Let r be the vmt vector in the direction of OP^ thus r = - = — .
p Xo
Let V be the unit vector perpendicular to the plane through OP and k\
positive in the direction of ^ increasing.
Let u be the unit vector perpendicular to OP in the above plane and
positive in the direction of 0 increasing.
Thus (r\r)=l = (u\u)^(v\v)', and (r|ti) = 0 = (r|t;) = (w|t;);
and u^ \ vr, v = | rw, r = j ttv.
(3) Also r = icos^sin ^ +Jsin0sin^ + A;cos^.
Hence v=rk = (% sin <f> — j cos tp) sin 0;
therefore t; ^j cos <]> — i sin <f>.
Ai^d u B |vr s i cos <f> cos 0 -h j sin <f> cos 0'- km.n0,
558 PURE VECTOR FORMULiK. [CHAP. IV.
(4) Again, . ^'='^^~ +^~oB"*"^
dp pidtl p sin 6d4> '
^ ^ \ Ip sin r9^/ ( i psmO J
^ \ \pd0J \ \psm0d(f>/ \ p / \ \pBinffJ p
(5) Again, let p be any vector function of the position of P, and let
where 7', v, u are the three rectangular unit vectors as defined above which
correspond to the position of P, and tTj, Wj, ^r,, are scalar functions of p, 0, ^.
Then (V \p) = (r | VisrO + (w l^«^2) + (v\^^n) + «^i(V |r) + w^(^ \u) + tsr,(V (if)
dfSTi disr^ BtBTg 2«ri isr. cot 0
= — f -I ? ^ — J — 1 .* ^ 1 .
dp pdd p sin 6d<l> p p
(6) Again, let q be the curl of p, and let g = /Cir + k^u + k^v.
Then | g = Vp = VisTif + Vot,u + VwjV + «riVr + «r,Vw + WaVt;.
Now Vr=:w- + v- = 0;
^ r V cos 0 Til
\u = — w. - + V : — 7, = — ;
p p sin u p
„ ( 7' sin 0 + i cos 6) (r sin tf -f w cos tf ) cottf 1
psinff p sin a p p
r 1 3«r. 1 acTs tsr, cot (9"|
Hence g = wv 1 — 3 5T + "" ^^ +
' ^ L p sin ^ 0^ p 3^ p J
tl 3«ri 9w, «rjl r 1 3cri 3«rj Wjl
sin ^ 9^ 9p p J L P ^^ ^P P J
--,, :3 5i?rji 1 9crj . cr, cot 0
Thus ^1 = - -5^ -^ — . — ^ ^rr 4-
p 9^ p sin ff d<p . p
^ 1 9«ri 9isr, tr
s
\
p sin d 90 9p p '
9tir2 1 9t!ri BTj
dp p d0 p '
364 Cylindrical Co-ordinates. (1) Employing cylindrical co-
ordinates, let fl?8 denote the length ON in the annexed figure, and <r deto^te
«t.
■s
SI
364]
CYLINDRICAL CO-ORDINATES.
559
the length NP. Also let v denote the same vector as in § 363, and w denote
a unit vector parallel to NP.
(2) Then v =j cos ^ — i sin ^, w = i cos (f> +j sin ^ ;
and w = \ vky t; = | kw, k = | wv.
Also
And
Hence
Again,
Fio. 2.
w
Vit; = — t; — = 0, V|w = t;
a<7» ^ <r«a(^» a^3« "^ <r a<r'
Vv = — t; - = — , \w = t; - = 0.
(3) Let any vector p be written WiW + «r,t; + israAr, and let its curl q be
written KiW + tc^v '\- kJc,
Then (V|p) = (V«r,|iI;) + (Vtsr,|t;) + (V«r3|i) + tsri(V w)-|-tir,(V |t;)
oiSTi , 1 0isr. 3t!r« Wi
3S — t ^ — — 2 ^ f ^ * ^
dvTf vr*.
-"[-l^-%•^Ty''[-^-MM^-^■
560 PURE VECTOR FORMULA. [CHAP. IV.
Hence 'fj=o oX + "~>
da a- d<f> a
1 OfSFt OVFo
^ 3a?8 3cr "
366. Orthogonal Curvilinear Co-ordinates*. (1) The formulae
may be generalized thus: let l, m, n be three unit rectangular vectors
associated with any point P, such that the system of vectors suffers a
continuous change in direction as the point P moves in a continuous line
to any other point P. Let Scti, S^Tq, Sct, be elements of arc traversed bj
the point as it moves through small distances from P in the directions
I, m, n respectively.
(2) Let P be determined by three curvilinear co-ordinates ^i, ^„ ^,,
such that during the small motion Scti, ^i becomes 6^ 4- S^i, and 0^ and 0t
are unaltered ; with two other similar specifications. Also assume that
^ HT * * ~ X" ' ~ XT '
where A,, A,, A, are functions of 0i, 02, 0^.
Then [cf. § 361 (2)] V = ig+m|| + ng.
(3) Thus if 0 be any scalar function of 0u 0%, 0ny
^» '■*=M(wf)+--^^<'">+ ®
Again, let p be any vector, and let p = «r,? + isTgrn + ^tsW..
Then
Similarly
^'^''[-w^w\'^'^iw'~'wr^'''\j0^
-h «riV? + «2^wi + BTsVyi (iii).
* Ab far as I am aware, the methods of transformation of the present and the two preoeding
articles have not been employed before. The methods are the analogue in this Algebra of Webb's
method of Veotor-DifFerentiation, published in the Messenger of Mathtmaties, 18S2, and ftillj
explained and applied to this ease in Jjove's Treatise on the Mathematiedl Theory of Elastieity,
J 119,
I
366] ORTHOGONAL CURVILINEAR CO-ORDINATES. 561
Thus when (V |Z), (V |m), (V |»), VI, Vm, Vw have been obtained, the
formulse for transformation are complete.
(4) Now 01 = constant, 0^ ^ constant, 0^ ^ constant, form three sets
of mutually orthogonal sur&ces.
1 1
Hence Z = | wn == ; V tfi = ^-5- \V0^V0^.
Hence (V |0 = rr ^ • V^, Vd, + V0^V0,V ^ .
Now V . V^aVtf, = W0^ . V^, - V^, . VVtf, = 0.
Therefore
Similarly
(V|m) = M^^^^. and (V I «) = A.M, ^ ;-\^ .
Hence from equation (ii) (V \p) = A.A^,|1- ^ + ^^ ^ + ^^ ^'^j .
(6) Again, V/ = V .^ V^, = vi. Vtf, = pVtf,VA,= i iVA,
d- •! I TT ^ ^^ Ai dA, , _ Ai dA, , At dAa
Similarly Vm = ,- gg^ m» - ^- gg- im. V„ = - _ „i - ^ ^-^^ „,„.
Hence from equation (iii)
+ AA0/'--- — -— — "^ w.
\9tfi A, 9^8 Ai/
Accordingly if q be the curl of p, that is | Vj^ ; and if g be written
Kil + /Cgm + /CjW,
*»-'^lao,A, a^, A,r
**""''*' Va^ A, " 3^, aJ-
w. 36
662 PURE VECTOR FORMULiE. [CHAP. IT.
These formulfle of course include as special cases the preceding' formalc
for polar and cylindrical co-ordinates [cf. §§ 363 and 364].
366. Volume, Surface, and Line Integrals. (1) Let dr stand for
an element of volume at the point x. Let dS be a vector-area representing
in magnitude and direction an element of surface at the point as. Then
dS =jkdS^ + kidS^ + %jd8, ;
where dSi^^^dx^^^ dS^^'dx^^, dSa^dxidx^.
Let \dS denote the normal, positive when drawn outwards.
Let (2a? be a vector line denoting in magnitude and direction an element
of arc at x. Then dx = idxi -i-jdx^ + kdx^, and dx^ is often denoted hy da.
(2) Then the well-known theorem connecting the volume and surfiice
integrals of any continuous function of position within a closed surfiEU^e is
jjj{V\u)dT^jj(udS),
(3) Green's Theorem can be written
jjj(y(f> I V^) dr = jj(<l>VyfrdS) --jjUvhIrdT
= fkf^<l>dS) - jjjitV^it>dT.
This can obviously be deduced from subsection (2) by writing 0 V-^ for «,
then
(4) Stokes' Theorem connecting line integrals and surface infcegrabis
expressed by
jj(yu\d8)=j{u\dx),
where the line integral is taken completely round any closed circuit, and the
surface integral is taken over any surface with its edge coincident with the
surface.
367. The Equations of Hydrodynamics. (1) Let the vector u
denote the velocity of a frictionless fluid at any point represented by the
vector X drawn from an arbitrarily chosen origin. Let p be the density at
that point, and «r the pressure. Let the vector / denote the external force
per unit mass at x. Also let the vector q denote the curl of w, so that
q =r |Vi^. The vector q defines the vortex motion at each point of the fluid:
portions of the fluid, for which 9 = 0 at each point, are moving irrotationally.
Then the fundamental equation of motion is
t'-^'*"^ »
+ (u|V)t. = -v(^ + V^) (ii).
366 — 368] THE EQUATIONS OF HYDRODYNAMICS. 563
(2) Assume in the first place that the fluid is homogeneous and in-
compressible : also that / is derivable from a force potential ^, so that
Equation (i) becomes
This can be transformed [cf. § 362 (7)] into
du
But by equation (iii) of § 362 (3)
Hence ^ + |gti = — V f — + '^ + iw*) (iii).
(3) The equation of continuity becomes
(V|w) = 0 (iv).
(4) These equations are independent of any special co-ordinate system.
Thus let d], d^y 09 denote any set of orthogonal curvilinear co-ordinates, so
that ^1 = constant, 0, = constant, ds = constant, denote three systems of
mutually orthogonal surfaces. Let Z, m, n and Aj, h^, h^ have the meanings
assigned to them in § 365.
Let u = Vil-\- v^m + t;,n, q = Kil-\- Kjn + «,n.
Then k^ = hji^ (sS" A* "" ^ i") ' ^^^ *w^ similar equations for k^
and Ki,
Now Z, m, n are independent of t. Hence equation (iii) splits up into
three equations of the type
dvi . hfi /«r
a« +*»•"-*•"' = - a^Ap
And equation (iv) becomes
1 i!L.+ 1 J^ + -L J^=. 0 (vi)
These are the general equations of motion of a homogeneous incom-
pressible fluid referred to any orthogonal curvilinear co-ordinates. They
include as special cases the equations referred to polar or to cylindrical
co-ordinates.
368. MoviNO Origin. (1) Equation (iii) of the preceding article
may be extended to the case of a moving origin. Suppose that the origin
moves with velocity v, then v may be a fonction of f, but of course is not a
function of position.
36—2
+ K^vz - '^'Va = - 9^( ~ + '^ + i^'j (v)-
664
PURE VECTOR FORMULA.
[chap. IV.
The point, which at time t is defined by the vector x, at the time t-hSt
is defined by the vector w — vSt.
Let ^ denote the stationary differential operator with respect to the
time relatively to the moving origin, so that k: gives the rate of change at
a moving point which is defined by the constant vector x drawn fi^m the
moving origin.
Then
0 9 , ,--v
r. = 5i + (HV).
^-(v\V)u + (u\V)u = -V(^ + ^y
St dt
Hence equation (iii) of § 367 becomes
^-(t»|V)«+|9u = -v(- + ^ + ^«) (vii).
(2) Equation (ii) of § 367 becomes
Now let tt' = tt — V. Then u' is the velocity of the fluid at any point
relatively to the origin.
Substitute u' + v for u and remember that (w | V) t; = 0 = (t? | V) w, since v is
not a function of x.
Also if V be the acceleration of the origin,
8i"St^^'
Hence the equation of motion becomes
^ +* + («' |V)«' = -V(-+^j (viii).
The equation of continuity is (V | «') = 0.
(3) The curl of u' is the same as that of u, since
|V« = (V («' + «) = I Vtt'.
Hence equation (viii) can be transformed into
Su'
+ 1> + |gf«' = - V (- + ^ + K") •
& - \p
Furthermore, since i is not a function of position, w = V (* |«).
Hence finally
+ !}«' = - v|-+^ + («|a!) + iu4 (ix).
(4) Therefore a uniform motion of the origin does not affect the form of
the hydrodynamical equation, when the velocity is reckoned relatively to the
origin.
St
369, 370] MOVING ORIGIN. 565
An acceleration of the origin adds a term to the force potential.
The vortices are the same whatever motion be assigned to the origin.
Therefore by suitable modifications of ^, equations (ii) and (iii) of § 367
may be looked on as the typical hydrodynamical equations, whether the
origin be at rest or be moving in any way.
369. Transformations of Hydrodynamical Equations. (1) Opera-
ting on (iii) of § 367 with V
dt
But V| =
dq
dt'
Also V|5M = (V|u)|5 + (V|u)|(y-(V|5r)|u-(V|5)|w = (w|V)|5r-(g|V)|w;
since (V |t^) = 0 = (V|5r).
Hence by taking the supplement
| + («|V)9 = (?|V)i*.
This can also be written
dq ^
dt
= (g|V)t^ (x).
(2) Again, operate on (iii) with V|.
Now ^lS = |<^l^)=^<>-
Hence Vqu = -V^i^ + '^'\'\vy\ (xi).
Also by § 362 equation (iv)
Vqu = {u IV) (V |tt) - mV»i* - (^uf = - uVhi - (Vw)».
370. Vector Potential of Velocity. (1) Assume that there are no
boundaries to the fluid which extends to infinity in all directions ; also that
the vortices [c£ § 367 (1)] only extend to a finite distance from the origin.
Now q = I Vtt.
Hence [cf. § 355 (2) equation (iv)]
Vg = V I Vu = I V (V |ia) - I V*M = - \V^.
Therefore by the ordinary theory of the potential, since by assumption
9 = 0 at all points beyond a certain finite distance from the origin,
\u
-^irjjj ^(x-a^y^''''
\
566 PURE VECTOR FORMULiE. [CHAP. IV.
where g' represents the curl of the velocity at the point «', and V stands for
o o o
i .r—, -f J 5— > 4- k ^— ? , and dr is an element of volume at x\
OOSi 0X2 vX^
(2) The integration may be assumed to be confined within any surface
large enough to contain all the vortices and such that none of them lie on
the surface.
Integrate by parts, and remember that by the assumption q' is zero
at all points of the surface.
Then [cf. § 362 (1)] |„ = ^///v' ^-.^dr'.
Now V ,. =-V
/v« •
^{x - xj ^(x - xy
Hence |« = ^/// V ^-^ dr' = ^V jjf -^^dr' (xii).
Hence -7- III ,, — r^ dr is a vector such that xi is its curl. Let this
AnrJJJ ^{x—xy
vector be denoted by p, then u = | Vp.
(3) Also by integrating by parts, it is easily seen that
since (^l9)=0.
The vector p is called the vector potential of the velocity.
(4) The same suppositions as to the absence of boundaries and as to
vorticity enable us similarly to solve equation (xi).
For by the ordinary theory of the potential equation (xi) can be trans-
formed into
- + i|r + iu« + constant = -.- \ll -77-^^^ dr.
p ^ ^ 4!7rJJJ v(^-^)
Now integrating by parts exactly as above,
= + tH-i.- + 7=iv///^(.^^*,' (.iii).
Hence if L denote the flux 1— 1 1 1 -n^ — yci dr, then
p OXi 0X2 uX^
■ J ■- l» Lg
371] VECTOR POTENTIAL OF VELOCITY. 567
871. OuRL Filaments of Constant Strength. (1) Let t; be any
vector which at each point is definitely associated with the fluid at that
point : the magnitude and direction of v may depend, wholly or in part, on
the velocity of the fluid and on its differential coefficients, and it may depend
partly on other properties of the fluid not here specified. Let it be assumed
that the components of v, namely Vi, t;,, t;,, and their differential coefficients
are single-valued and continuous functions. Let r denote the curl of v, so
that r = I Vt?.
Lines formed by continually moving in the direction of the r of the
point are called the curl lines of the vector v. Since r fulfils the solenoidal
condition, namely (V|r) = 0, such lines must either be closed or must begin
and end on a boundary.
(2) A curl filament is formed by drawing the curl lines through every
point of any small circuit in the fluid. If dS be the vector area at any point
of a curl filament, then rdS is called the strength of the filament. It follows
from the solenoidal condition by a well-known proposition that the strength
of a given curl filament is the same at all points of it.
Let any finite circuit be filled in with any surface, then by § 366 (4),
jj{rdS)=j(v\dx)i
where the line integral is taken round the circuit.
(3) Let us now find the condition that the sum of the strengths of the
filaments, which pass through any circuit consisting of given particles of the
fluid, may be independent of the time. Also for the sake of brevity assume
that the region of space considered is not multiply connected.
The condition is ;// K^ '^^ ~ ^'
This becomes [(- \d^ +j(v\du)^ 0.
Now du = {dx I V) t^.
Therefore (v \du) = (da? | V) (v\u) = V(v \u) \dx.
Hence j(^^\dx^+ j {v\du) = 0,
becomes / jdl "^ ^ ^^ | w)i | da? == 0.
Now if -^ be any scalar function of x and t which together with its
differential coefficient is continuous and single-valued.
J(VVr|da?) = 0,
I
568
PURE VECTOR FORMULA
[CHJLF. IV.
where the integration is completely round the circuit. Hence if '^ be some
such scalar function, we deduce
dv 3» , , |_.
(4) Also
Now
(m I V) t> = V (v |u) + Vv |u = V (i; |u) - |u . Vi; = V (v I u) - 1 . t^ I Vt; = V (t; |tt) — I tir.
Hence
Now put
Then
||^|^,r + V(^;|u) = -V^.
X = ir + (v\u).
dv
dt
+ |ru = — Vj^
.(xiv).
(5) To eliminate x we operate with V, then
| + V|r«=0.
Now V |ur = (V |r)|w — (V \u)\r.
But (V|r) = 0 = (V|tt)|.
Hence V |t^r = (r | V) |w — (t^ | V) \r.
Therefore the equation becomes after taking supplements,
dr
This is
g^ + (t*|V)r = (r|V)w.
.(XV).
(6) This condition should be compared with equation (x). It follows
from the comparison that the strengths of all vortex filaments are constant.
In other words, that if equation (xiv) be conceived as an equation to find the
unknown vector r, then q is one solution for r. But q is not necessarily the
most general solution. Thus there are other curl filaments in the fluid with
the same property of constancy.
(7) But equations (xiv) and (xv) are more general than these enunciations
would suggest. For in the derivation of (xiv) neither the equation of con-
tinuity for an incompressible substance nor the kinetic equation of fluid
motion were used. It follows that if the motion of any continuous substance
be assumed given, so that ti is a given function of x and t, then any vector v,
as defined in subsection (1), with its curl r which satisfies equation (xiv) is
such that the curl filaments are of constant strength.
(8) Equation (xv) involves the equation (V |m) = 0. Hence this equation
holds for any incompressible substance moving in any continuous manner.
^
372] CURL FILAMENTS OF CONSTANT STRENGTH. 669
An extended form of (xv) can be deduced by writing (V |t4) = tf, where 0
is a known function of x and t^ since u is such a function.
Hence V \ur = (r | V) \u - (u | V) |r - tf |r.
Therefore ^ + (w | V) r = (r | V) t^ - tfr ;
that is, t: = (r|V)tt — flr (xvi).
372. Carried Functions. (1) Let ^ be a scalar function of x and t
such that for all values of t any surface ^ = 7, where 7 is any pai-ticular
constant, represents the same sheet of particles of the substance. Then the
function <f> will be called a carried function*.
(2) The analytical condition which <f> must satisfy is
or as it may be written,
dt "'
g^+(«.|V^) = 0.
Also |;V0 = |-V0 + (u|V)V^ = V^ + V(u|V)^
dt ^ dt ^ ^ ^ ' ^ dt
= V^-V(t*|V)^ + V(u|V)^ = -V(ti|V)^ + V(tt|V)0
= -V(V^|w) (xvii).
(3) Now let if> and '^ be any two carried functions. Then by equation
(xvii)
J ^_
^ ( V^ V-^) = - V (V^ I tt) V^ - V^ V (V^ 1 m)
= -V{(V^|tt)Vi^-(V^|u)V^} = -V{V0V^|tt}...(xvm);
by § 355 (2) equation (iii).
(4) Also if 0, -^j ^ be any three carried functions
V{(V^VtVx)l«},
by the extended rule of the middle factor, where the product of three vectors
is treated as an extensive magnitude formed by progressive multiplication.
Hence
^(V^vV'Vx) = - (V^v^rVxXV ;«) — ^(V^vvrVx)...(xix).
* These functions for a perfeot flnid have been inyestigated by Clebsoh in Crelle, Bd. lyi. 1860,
and by M. J. M. Hill, in the Transactions of the Cambridge Philosophical Society, Vol. xiv. 1888.
670 PURE VECTOR FORMULiE. [CHAJP. IV.
This result is obtained by Hill, without the use of the Calculus of
Extension, in the paper cited. The brevity of the necessary analysis hy this
method is to be noted.
(6) Putting S for the determinant (V^V-^V;^), equation (xix) can be
written
and it follows at once that
g = -(^-.^)g, ^ = -(&-3tfd+^)S, andsoon.
Hence if none of the series 0, u, 0, and so on, are infinite, then all the
successive mobile differential coefficients of S with regard to the time are
zero when S is zero.
Hence if S is zero at each point at any one instant, it remains zero at all
subsequent times.
373.. Clebsch's Transformations. (1) The curl filaments, defined by
§ 371, equation (xvi), move with the substance with unaltered strength.
Let two systems of sur&ces be drawn at any instant on which the curl lines
lie. Then if these surfaces be defined at any instant by the carried functions
<l> and ylr, the intersections of the two systems at any subsequent instant will
define the curl lines.
Therefore, remembering that V^ and Vy^ are vectors at each point
respectively perpendicular to the surfaces ^ = constant, and -^ = constant,
passing through that point, we may write r = | V^V-^ = «■ IV^V-^, where «• is
some function of x and t.
(2) But from equation (xvi), -it = (r , V) w — ^.
dv
Now 31 = «■ I V^V-^ + «r
dt
dt
V^V-^zr: w \Vif>V^ - «r |V jV^V^Ia}.
dt
Also (r I V) w - ^ = w {(V0 Vi|rV) u - 1 V^Vi|r (V \u)]
= tsr I {(V0 V-^ V) I u - V^V-^ (V I u)]
= tir I {Vi|r V ( V<^ I tt) + VV<^ (V^ I m)}
= tsr I V {- V-^ (V0 I w) + V0 (Vi|r |ii)}
By equating these results we obtain tar | V^V-^ = 0.
But by hypothesis the vector [V^V-^ is not null. Therefore «r = 0.
Hence w is a carried function of the substance.
373] CLEBSCH*8 TRANSFORMATION& 571
Let w be replaced by the carried function x- Thus
r = j^ I V^^'^, and I r = x^^V-^.
(3) Now the solenoidal condition (V | r) = 0, gives
V . x^i>^'ir = 0, that is V;;^V0V'^ = 0,
since V . V(f>V^ft = VV^ . V-^ - V^ . W-^ = 0.
But Vj^V^V-^ is the well-known Jacobian whose vanishing is the con-
dition that X is a function of ^ and y^, where t is regaixled as a constant.
Hence X=/(<^, ^, 0-
But since if>, y^^ x ^^^ carried functions, -^ = ^ = 0 ; where ^ means
that if> and y^ are regarded as constant. Hence x^^ ^ function of (j) and y^
only, where t is regarded as a variable. Thus x —f(^> '^X
(4) It is now easy to prove that the most general form for these curl
filaments, which satisfy equation (xvi), is
r = |V^V'^ (xx).
For let AT be a carried function of the form /(<}>, ^). Then ^ and r-r-
are carried functions of the same form. Then we have proved that the most
general form for r is ^-r IV^V^.
But v. = ^V<^4-^V^.
dur
Hence VsrV-^ = ^-r V<j>Vyft.
Thus the most general form can be converted into jVwV-^, which is the
form stated in equation (xx).
(5) Now V<^V^ = V . <^Vi|r = - V . i|rV^.
Hence firom the preceding subsections of this article the most general
form of the solution of equation (xvi) for the vector v, of which r. is the
curl, is given by
v^^V^ + Vcr (xxi),
where <}> and y^ are carried functions, and ^ is any continuous function of x
and t
(6) We can also solve for the function x which appears in equation (xiv)
in terms of ^, ^ and m. It is to be noted that the x ^^ equation (xiv) is not
a carried function.
572 PURE VECTOR FORHUUB. [CHA-P. IV.
Now by equation (xx)
\ru = V^V^ |u = (tt |V^) V^ - (u |V^) V^ = _|^ V^ + ^ V^ ;
since [cf. § 272 (2)]
Also ^ = I K^^t) + Vw} = ^^t + 4>'^^t + ^^t.
Hence equation (xiv) can be written
Therefore ^ {x + H^t + tire} = 0.
Now only the differential coefficients of x s^pp^ar in equation (xiv), so we
may with perfect generality write
X='-<M^t'-vt (xxii).
Equations (xxi) and (xxii) are the extension of Clebsch's transformations
for the velocity of a perfect fluid.
374 Flow of a Vector. (1) The flow of a vector v along any
unclosed curve will be defined to be the integral
/'
(v \dx),
where the lower limit is the starting point of the line, curved or straight,
and the upper limit is the end point.
(2) If the vector v be such that its curl filaments are of constant
strength, then its flow between any two points P and Q along a defined line
takes by equation (xxi) the form
[Q
(3) The part I ifxly^ is such that it is independent of the time if the
same line of particles be always considered. But it does in general depend
on the special line of particles chosen, and is not completely defined by the
terminal particles.
The part wq - ^p is completely defined by the terminal particles, but
varies with the time.
(4) Suppose that the mobile differential coefficient of the flow of any
vector V along any line of particles in the substance is always equal to the
374] PLOW OF A VECTOR. 573
flow of some vector p along the same line of particles, then p will be called
the motive vector of the flow of v.
The definition of jp is therefore ^\(v \dx)= Up \dx).
By attending to the derivation of equation (xiv) it is easy to see by the
use of the same analysis as there employed that
jp = ^ + |rM+VOT (xxiii),
where v is some single-valued scalar Amotion of x and ^, r is the curl of v^
u is the velocity of the substance at the point as.
This equation should be compared with the equations of Electromotive
Force in Clerk Maxwell's Electricity and Magnetism, Vol. ii., Article 598.
Note. The present chapter is written to shew that formula and methods which have
been developed by Hamilton and Tait for Quaternions are equally applicable to the
Calculus of Extension. The pure vector formulsB have some affinity to those of the very
interesting algebra developed by Prof. J. W. Gibbs, of Yale, U.S.A., and called by
him Vector Analysis. Unfortunately the pamphlet called, * Elements of Vector Analysis,'
New Haven, 1881 — 4, in which he developed the algebra, is not published, and theiiefore is
not generally accessible to students. The algebra is explained and used by Oliver Heaviside,
loc, cit. p. 550 ; it will be noticed in its place among the Linear Algebras.
Note on Gbassmann.
H. Qrassmann's Atudehnungslehre von 1844 was republished by him in 1878 (Otto
Wigand, Leipzig).
A note by the publisher in this edition states that the author died while the work
was passing through the press. A complete edition of Grassmann's Mathematical and
Physical Works (he also wrote important papers on Comparative Philology) with
admirable notes is now being published under the auspices of the Boyal Saxon Academy
of Sciences, edited by F. Engel (Leipzig, Teubner) Band i. Theil i. 1894, Band i. Theil ii.
1896 ; the remaining parts are not yet published (December 1897). I have not been able
to make any substantial use of this admirable edition : the present work has been many
years in composition and already nearly two years in the press ; and the parts most closely
connected with Grassmann's own work were, for the most part, the first written.
It must be distinctly imderstood that the present work does not pretend to exhaust the
suggestions in Grassmann's two versions of the ^Ausdehnungslehre' : I only deal with those
parts, which I have been able to develope and to bring under one dominant idea. Thus
Grassmann's important contribution to the theory of Pfafifs Equation by the use of the
Calculus of Extension, given in the Atudehnungslehre of 1862, is not touched upon here.
It is explained in Forsyth's work. Theory of Differential Equations, Part i. Chapter v.
The following list of the mathematical papers of Grassmann is taken from the
Royal Society Catalogue of Scientific Papers.
574 NOTE ON ORASSMANN.
Theorie der Ceniralen, OreUe xxiv. 1842 ; and xxv. 1843.
Ueber die Wissenschaft der extensiven Grttsse oder die Auadehnungalelires,
Arehiv vi. 1845.
Neue Theorie der Electrodynamik, Poggend, Annal. lxiv. 1845.
QrimdzUge zu einer rein geometriachen Theorie der Curven, mit Anwendixng' einer rein
geometrischen Analyse, Crdle xxxi. 1846.
G^metrische Analyse gekniipft an die von Ijeibnitz erfundene geometruaclie Ohanscfe-
ristik, Leipzig^ JaUon. Premchr. (No. 1) 1847.
Ueber die Erzeugung der Gurven dritter Ordnung durch gerade Linieriy und Uber
geometnsche Definitionen dieser Ciiryen, CreUe xxxvi. 1848.
Der allgemeine Satz tiber die £rzeugung aller algebraischer Curven durcli "Bg^^^ui^
gerader Linien, Crelle xui. 1851.
Die h5here Prqjectivitat und Porspectivitat in der Ebene; dargestellt durch geo-
metrische Analyse, Ordle xlii. 1851.
Die hOhere Projectivitat in der Ebene, dargestellt diurch FiinctionsverknUpfungeD,
CreUe xlii. 1851.
Erzeugung der Curven vierter Ordniuig durch Bewegung gerader Linien, Crdle
XLiv. 1852.
Zur Theorie der Farbenmischung, Poggend, Annal. lxxxix. 1853; and Phil. M<ig,
xn. 1854.
Allgemeiner Satz iiber die lineale Erzeugung aller algebraischer Oberfiachen, Ordle
XLix. 1855.
Grundsatze der stereometrischen Multiplication, CrdU xlix. 1855.
Ueber die verschiedenen Arten der linealen Erzeugung algebraischer Oberfiachen,
CrdU xiiix. 1855.
Die stereometrische Gleichung zweiten Grades, und die dadurch dargestellten Ober-
flSchen, CreUe xlix. 1855.
Die stereometrischen Gleichungen dritten Grades, und die dadurch erzeugten Ober-
fliichen, CreUe xlix. 1855.
Sur les diff^rents genres de multiplication, Crdle xlix. 1855.
Die lineale Erzeugimg von Curven dritter Ordnung, CreUe Lii. 1856.
Ueber eine neue Eigenschaft der Steiner'schen Gegenpunkte des Pascal'schen Sechs-
ecks, Crdle Lvin. 1861.
Bildung rationaler Dreiecka Angenaherte Construction von ir, Arehiv Math. Phys.
XLIX. 1869.
LOsung der Gleichung a:*4-y* + «8 + it' = 0 in ganzen Zahlen, Arehiv Math. Php,
XLIX. 1869.
Elementare Aufl5sung der allgemeinen Gleichung vierten Grades, Arehiv Math. Phft,
LI. 1870.
Zur Theorie der Curven dritter Ordnung, O&ttingen Nachrichten^ 1872.
Ueber zusammengehOrige Pole und ihre Darstellung durch Producte, GiftHngen
Nachrichteuy 1872.
Die neuere Algebra und die Ausdehnungslehre, MatL Annal, vii. 1874.
Zur Elektrodynamik, Crdle Lxxxm. 1877.
Die Mechanik nach den Principien der Ausdehnungslehre, Malh, Annal, xn. 1877.
Der Ort der Hamilton'sohen Quatemionen in der Ausdehnungslehre, Math, Annul,
xn. 1877.
Yerwendung der Ausdehnungslehre fur die allgemeine Theorie der Polaren und
den Zusammenhang algebraischer Gebilde (posthumous), Crdle lxxxiv. 1878.
An obituary notice will be found in the Zeitschrift Math, Phye. Vol. xxm. 1878,
by Prof. F. Junghans of Stettin.
NOTE ON ORASSMANN. 575
The works on the Calculus of Extension hj other authors deal chiefly with the
application of the Calculus to Euclidean Space of three dimensions, to the Theory of
Determinants, and to the Theory of Invariants and Covariants in ordinary Algebra.
Thus they hardly cover the same groimd as the parts of the present work, dealing with
Grafismann's Calculus, except so feur as all are immediately, or almost immediately,
derived from Giueemann's own work. Some important and interesting works have been
written, among them are:
Abriss des geometrischen KcdkiUsj by F. Kraft, Leipzig 1893 (Teubner).
Die Auidehnungdehri oder die WiueMcho^t von den extensiven OrHuen in strenger
F^ormel-Entwicklungj by Robert Grassmann, Stettin 1891.
Sjfatem. der Raundehre, by V. Schlegel, Part 1. 1872, Part II. 1875, Leipzig (Teubner).
Calcolo QeometncOy by G. Peano, 1888, Turin (Fratelli Bocoa).
Iniroduetion d la Q^omArie Diff4refntieUe^ euivant la MMode de ff, Orasemann^ by
C. Burali-Forti, 1897, Paris (Gauthier-Villars et fils).
The Directional Calctdua^ by E. W. Hyde, 1890, Boston (Ginn and Co.).
I did not see the above-mentioned work by C. Burali-Forti till the whole of the
present volume was in print. It deals with the theory of Vectors and of Curves and
Surfaces in Euclidean Space, in a similar way to that in which they are here dealt with
in Chapters i., m., and iv. § 359 of Book vii. The operation of taking the Vector is
explained and defined. The formulsB of multiplication in so far as they involve supple-
ments are however pure vector formulsB : some interesting investigations are given which I
should like to have included : the application to Gauss' method of curvilinear co-ordinates
is also pointed out.
Buchheim's and Homersham Cox's important papers have already been mentioned
[cf. notes pp. 248, 370]. I find that Buchheim has already proved [c£ Proc. Lond. Math.
Soc. VoL xvin.] the properties of skew matrices of § 155 : also the extension of the idea
of Supplements in Chapter iii. Book iv. is to some extent the same as his idea of taking
the polar {cf. Proc. of Lond. Math. Soc. VoL xvi.]. I had not noticed this, when writing
the above chapter. He does not use the idea of * normal intensity'; accordingly his
point of view is rather different. He does not bring out the fundamental identity of
his process of taking the polar with Grassmann's process of taking the supplement.
Homersham Cox has also written a paper*, Application of Orassmann*$ Auedehnunge-
lehre to Properties of OirdeSj Quarterly Journal of Mathematics, October, 1890.
There are two papers by E. Lasker, An Eesay on the Oeometrieal Calculus j Proc. of the
Loudon Math. Soc. VoL xxviil 1896 and 1897. The paper applies the Calculus to
Euclidean space of n dimensions and to point-groups in such a space. It contains results
which I should like to have used, if I had seen it in tima
Heknholtz uses Grassmann's Calculus, as far as concerns addition, in his ffandbuch
der physiologischen Optik, § 20, pp. 327 to 330 (2nd Edition).
* In this paper by a slip of the pen the words 'Outer' and 'Inner' as applied to maltiplication
are interehanged.
INDEX.
The references are to pages.
Abflolnte in oonneotion with Gongrnent Trans-
formations, 456 »qq,y 500 »qq, ;
Conio Section, defined, 497 ;
Plane, 867, 496 999.;
Point-pair, defined, 351;
Points on, in Hyperbolic Geometry, 422 ;
Polar Regions, 367, 384, 420;
Qnadrio, defined, 855.
Absorption, Law of, 86 aqq.
Acceleration in Endidean Space, 540 aqq. ;
in Non-EacUdean Space, 482 sqq.
Addition in connection with Classification of
Algebras, 29 sqq, ;
and Maltiplioation, 25 sqq, ;
and Positional Manifolds, 120 sqq,;
in Algebra of Symbolic Logic, 35 aqq,;
of Vectors, 507 »qq, ;
Principles of, in Uniyereal Algebra, de-
fined, 19 sqq, ;
Belations, 123.
Algebra, Linear, mentioned, 32, 172;
and Algebra of Symbolic Logic, 35 ;
defined, 28;
Universal, mentioned, 11, 35 ;
defined, 18.
Algebras, Classification of, 29 sqq. ;
Linear Assooiatiye, 80 ;
Species of, defined, 27 ;
Nnmerical Genas of, 29, 119.
Angle of Contingence, in Endidean Space, 539
and 552;
in Non-Endidean Space, 479 ;
of Parallelism, in Hyperbolic Space, 488 ;
of Torsion, in Endidean Space, 540 ;
in Non-Endidean Space, 479.
Angular Distance between Points in Anti-space,
417.
Anharmonic Batio in Positional Manifolds,
defined, 132;
of Systems of Forces, 290.
W.
Antipodal Elements, defined, 166;
in Elliptic Space, 361;
Form of Elliptic Geometry, defined,
355;
Intercept, defined, 168;
length of, 362.
Anti-space, considered, 414 sqq. ;
defined, 354.
Anti-spatial Elements, 414 sqq.
Arbitrary Regions in Algebra of Symbolic Logic,
55 sqq.
Arithmetic and Algebra, 11.
Associated Quadric of Triple Gronp, see Gronp;
System of Forces, see System of Forces.
Associative Law and Algebra of Symbolic Logic,
37;
and Combinatorial Multiplication, 174 ;
and Matrices, 251 ;
and Multiplication, 27 ;
and Pure and Mixed Products, 185 ;
and Steps, 25 ;
defined, 21.
Ansddinungslehre, 13, 19, 32, 115, 131, 168,
171, 172, 180, 198, 210, 219, 229, 248, 262,
278, 317, 522, 573.
Axis, Central, in Elliptic Geometry, 401 ;
in Euclidean Geometry, 529 ;
in Hyperbolic Geometry, 454;
of a Congruent Transformation in Elliptic
Space, 471;
in Euclidean Space, 501 ;
in Hyperbolic Space, 458 ;
of a Dual Group, see Group.
Ball, Sir R. S., 281, 870, 406, 462, 473, 475,
581, 532.
Base Point, 518.
Beltrami, 869, 451.
Binomial Expressions in Symbolic Logic,
45 sqq,
37
578
INDEX.
Biquaternions, 870, 898.
Bolyai, J., 869, 426, 436, 451, 487;
Wolfgang, 369.
Boole, 4, 10, 35, 46, 68, 96, 111, 115, 116.
Bradley, 6, 10.
Brianchon*s Theorem, 231.
Buchheim, 248, 253, 254, 278, 370, 405,
675.
BuraU-Forti, 622, 576.
Bumside, 370.
Calculus, General Nature of, defined and dis-
cuBsed, 4 $qq. ;
Differential, distinguished from Uni-
versal Algebra, 18 sqq, ;
of Extension, Algebraic Species of, 28, 31;
and Descriptive Geometry, 132,
214 sqq, ;
and Theory of Duality, 146, 196,
212, 481 ;
investigated, 169 sqq.
Cantor, G., 16.
Carried Functions, 569 sqq,
Cayley, 119, 131, 135, 161, 248, 249, 851, 352,
353, 354, 369.
Central Axis, see Axis.
Central Plane of Sphere in Hyperbolic Geo-
metry, 442.
Centre of Dual Group, see Group.
Characteristic Lines of Congruent Transforma-
tion, 470.
Characteristics of a Scheme, 9, 14 ;
of a Manifold, 13.
Chasles, 246.
Circle, defined, and Perimeter of, in Elliptic
Geometry, 875 ;
Great, on Sphere in Hyperbolic Geo-
metry, 448 ;
in Hyperbolic Geometry, 484.
Classification (operation in Symbolic Logic),
41.
aebsoh, 869, 669, 570;
and Lindemann, 278, 280, 294.
Clifford, 13, 869, 870, 398, 406, 407, 409, 472.
Combination, General Definition of, 8.
Common Null Line of a Group, 286.
Commutative Law and Algebra of Symbolic
Logic, 87 ;
and Congruent Transforma-
tions, 464 ;
and Multiplication, 27 ;
defined, 21.
Complete Manifold, defined, 16.
Complex Intensity, 120.
Complexes, Conjunctive flu&cl
fined and discussed, 107 «99- ;
Linear, 278 sqq.
Compound Extensive Magziitades, defiTifd, 17i
sqq.
Congruence of Terms, defined, 122.
Congruences of Lines and I>iuJ Oxtnips, 28S.
Congruent Banges, defined, 31^ ;
Klein's Theorem concerning, 353 ;
Transformations in Kuolidean Geomclzj,
500 sqq,f 586 sqq. ;
in Non-Euclidean Qeom0Uy,4Si
sqq.\
and Work, 469, 477, 537 ;
Associated Systems of Fcnoes of,
466, 476, 536 ;
Parameters of, 460, 471.
Conies, Descriptive Geometry of, 229 9qq.
Conjugate Co-ordinates, defined and diaeaaaed,
14S sqq.;
Lines, 277 ;
Sets of Systems of Forces, 298, 806, SOB.
Conjunctive Complex, 107 sqq.
Construction, Grassmann*8, 219 sqq. ;
in a Positional Manifold, defined, 214 ;
Linear, of Cubics, 233 sqq» ;
von Staudt's, 215 sqq.
Content, Theory of, 370, 406, 462.
Contingence, Angle of, see Angle.
Co-ordinate Elements, defined, 125 ;
Region, defined, 126.
Co-ordinates, Curvilinear, in Eadidesn Space,
543;
in Non-EuoUdean Space, 488, 494;
Conjugate, defined, 148.
Cox, Homersham, 346, 370, 399, 400, 401, 576.
Cubics, Linear Construction of, 283 sqq.
Curl, defined, 664 ;
Flux, defined, 554 ;
Lines and Filaments, defined, 567.
Curvature of Curves and Surfaces in Eueiidean
Space, 544 sqq. ;
in Non-Euclidean Space, 479 sqq.
Curve-Locus in Space of r Dimensions, defined,
180;
Quadriquadric, 144, 151.
Curvilinear Locus in Space of p Dimensioii^
defined, 180.
Cylindroid in Euclidean Space, 582 ;
in Non-Eudlidean Space, 403, 455.
De Morgan, 82, 100, 122, 181.
Dependence in a Positional Manifold, dflfin^
123.
1
INDEX.
679
IXerivation, General Definition of, 8.
I>e(erminant8 and Combinatorial Multiplica-
tion, 180;
asBooiated with Matrices, 252.
Determining Property of a Scheme, 8.
Developable Surface in Non-Euclidean Geome-
try, 481.
Development in the Algebra of Symbolic Logic,
45 8qq,
Diametral Plane of a Daal Group in Euclidean
Space, 532.
Dimensions of a Manifold, defined, 17.
Direct Transformation of a Quadric, 338 sqq,;
of the Absolute, 456 $qq.
Director Force, Ijine, or Equation, of a Group,
286 sqq.
Discourse, Umyerse of, 100.
Discriminants of Equations, defined and dis-
cussed, 51 sqq,;
of Subsumptions, defined, 59.
Disjunctive Complex, 107 tqq.
Displacements of Rigid Bodies in Euclidean
Space, 500 tqq,, 536 ;
in Non-Euclidean Space, 456 tqq,;
Small, 464, 476;
Surfaces of Equal, 462, 472;
Vector, 472 tqq.
Distance, General Theory of, 349 tqq,\
in Elliptic Geometry, Shortest, 385, 387
tqq,;
in Hyperbolic Geometry, 416 tqq,;
Angular between Points, 417;
between Planes, 428;
Shortest, 429 tqq,;
in Non-Euclidean Geometry between
Sub-regions, 365 tqq,;
Definition of, 352.
Distributiye Law in Algebra of Symbolic Logic,
37, 84, 174.
Divergence, defined, 554.
Division in Algebra of Symbolic Logic, 80 tqq, ;
of Space in Non-Euclidean Geometiy,
355, 379.
Dual Group, tee Group.
Duality, Theory of, 147, 196, 481.
Dupm's Theorem in Euclidean Space, 546;
in Non-Euclidean Space, 494.
Elements and Terms, 21 ;
Antipodal, 166, 361;
Co-ordinate, 125;
Intensively Imaginary, 166;
Beal, 166;
Linear, defined, 177 ;
Elements, Linear, in Euclidean Space, 508 tqq, ;
in Non-Euclidean Space, 399 tqq,,
A52eqq.;
in Positional Manifold, and Me-
chanical Forces, 273;
Null, in Symbolic Logic, 35, 37, 38;
and Prepositional Interpretation,
109, 111 ;
in Universal Algebra, 24 tqq,, 28;
Planar, defined, 177 ;
of a Manifold, defined, 13 ;
Secondary Properties of, 14;
Regional, defined, 177;
Self-Normal, 204 tqq.;
Spatial and Antispatial, in Hyperbolic
Space, 414 tqq,;
Supplementary, in Symbolic Logic, 36
tqq.
Elimination and Syllogisms, 103 tqq. ;
from Existential Expressions, 89 tqq,;
in Symbolic Logic, defined and discuss-
ed, 47 tqq,;
Formula for, 55.
Elliptic Definition of Distance, 352 ;
Space, Formulfe for, 356 tqq,;
Kinematics of, 470 tqq,;
Parallel Lines in, 404, 407 tqq,;
Subregions in, 397 tqq.;
Vector Systems of Forces in, 406 tqq. ;
Transformations in, 472 tqq,;
Spatial Manifold, defined, 355.
Engel, F., 317, 369, 370, 573.
Equal Displacement, Surface of, 462, 472.
Equations, Identical of Matrices, 256, 261 ;
in Extensive and Potitional Manifoldt,
viz.
Defining, defined, 162;
Director, of Groups, 286;
of Condition, defined, 172;
of Subregions, 195;
Plane and Point, 147;
Reciprocal, defined, 147 ;
in Algebra of Symbolic Logic, viz.
and Universal Propositions, 105;
Auxiliary, 71, 78;
Limiting and Unlimiting, defined, 59;
Negative and Positive Constituents of,
defined, 50;
Simultaneous, 51 tqq, ;
Discriminants of, defined, 52;
Resultant of, defined, 52;
with many Unknowns, 52;
Discriminants of, 53 ;
Resultants of, 53;
580
INDEX.
Eqaations, with many Uuknowns, Solation,
65 $qq»;
Johnson's Method, 73 aqq.;
Skew-Symmetrical, 71 sqq,;
Symmetrical, 67 sqq., 73 fqq.^
75 sqq.;
with one Unknown, 49 sqq.;
Discriminants of, 51 ;
Besultant of, 51 ;
Solation of, 55 ;
Standard Form of, 49.
Equivalence, defined, 5;
Definition concerning, in Universal Al-
gebra, 18;
in Symbolic Logic, interpreted, 38 ;
for Propositions, 108 ;
Proof of, 36.
Euler's Theorem, in Euclidean Qeometiy, 547 ;
in Non-Endidean Geometry, 492.
Existential Expressions, 83 sqq,;
and Prepositional Interpretation,
111 sqq,;
Besultant of, defined, 90;
Solution of, 91.
Expressions, Field of, defined, 60.
Extension, Calculus of, investigated, 169 sqq,;
mentioned, 28, 31, 132, 146;
of Field, maximum and minimum, de-
fined, 61.
Extensive Magnitudes, defined, 176 sqq.;
Manifolds and Non-Euclidean Geometry,
899 sqq,, 452 sqq,;
defined, 177;
of three dimensions, 273 sqq,
Extraordinaries, defined, 119.
Field, Limited, defined, 61;
of an Expression, defined, 60;
of an Unknown, defined, 60.
Flow of a Vector, defined, 572.
Flux, defined for Point and Vector FormulsB, 527;
defined for Pure Vector Formulie, 549 ;
Multiplication, 528 sqq.;
Operation of Taking the, 522, 527.
Flye, Ste Marie, 369.
Force, defined, 177;
Compared with Mechanical Force, 273 ;
Single, Condition for, 277.
Forces, Director, of a Group, 286;
Groups of Systems of, see Groups;
Intensity of, in Euclidean Space, 525;
in Non-Euclidean Space, 899, 452;
Investigated, 278 sqq,;
Spatial, investigated, 452 sqq.;
Forces, Systems of, see Systems.
Forq4h, 17, 573.
Franklin, Mrs, 98, 116.
Frischauf, 869.
Functions, Theozy of, 11.
Gauss, 488, 490, 543.
Generating Begions of Quadrios, 147 «99*« 153
sqq,;
in Non-Euclidean Geometiy,
397, 451.
Generators, Positive and Negative SysteanB of,
207.
Geometry and Algebra, 11;
and Extensive Manifolds, 273 ;
Descriptive, and Calculus of Bxteosion,
214;
of many Dimensions, 131 ;
Elliptic, Polar and Antipodal Forms,
defined, 355;
Hyperbolic, Investigated, 414 sqg,;
Line-, 278;
Non-Euclidean, and Cayley's Theory oi
Distance, 351;
and Lindemann's Theozy of Forces,
281;
Historical Note upon, 869 9qq,;
of a Sphere, Euclidean, 365 ;
Parabolic, 496 sqq,;
as a limiting Form, 367 ;
Spherical, 355.
Gerard, 869.
Gibbs, 573.
Graphic Statics, 520.
Grassmann, H., 13, 19, 28, 31, 32, 115, 122,
131, 132, 146, 168, 171, 172, 201, 210, 219,
229, 233, 235, 246, 248, 249, 262, 278, S70,
522.
Grassmann, H., Note upon, 573 sqq.^
H. (The Younger), 317;
B., 575.
Graveltus, H., 281.
Green's Theorem, 562.
Groups of Systems, 284 sqq,;
Common Null Lines of, 286 ;
Dureotor Forces of, 286 ;
Dual and Quadruple, 287 sqq,;
and Congruences, 288 ;
Central Systems of, 402, 455;
Centres of, 402;
Diametral Plane of, 582 ;
Elliptic and Hyperboliev deAned,
292;
Parabolic, 289, 296 sqq.;
I
INDEX.
581
Groups, Dual, in Elliptio Space, 402 sqq, ;
in Eadidean Space, 531 sqq, ;
in Hyperbolic Space, 455 ;
Principal and Secondary Axes of, 582;
Self-Snpplementary, 292, 296 sqq, ;
Invariants of, see Invariants ;
Qnintnple, 286 sqq.;
Beciprooal, 285 sqq.;
Semi-latent and Latent, defined, 822 ;
Types of, 826499.;
Triple, 295 sqq, ;
Associated Qaadrio of, 295, 314,
535;
Conjugate Sets of Systems in, 298,
306 sqq, ;
in Eoelidean Space, 538 sqq.
Halsted, 369.
Hamilton, W. B., 32, 115, 131, 552, 578.
Hamiltonian, defined, 554.
Hamilton's Differential Operator, 554.
Hankel, 32.
Harmonic Invariant, see Invariant.
Heaviside, Oliver, 550, 578.
Helmholtz, 168, 369, 575.
Hill, M. J. M., 569, 570.
Homography of Banges, defined, 133.
Houel, 369.
Hyde, E. W., 575.
Hydrodynamics, 562 sqq.
Hyperbolic Definition of Distance, 352 ;
Dual Groap, see Group;
Geometry, Formulas for, 362 sqq, ;
investigated, 414 sqq.;
Spatial Manifold, defined, 855.
Ideal Space, 414.
Identical Equation of a Matrix, 256, 261.
Incident Regions in Algebra of Symbolic Logic,
42;
in Positional Manifolds, 125.
Independence of Elements in Positional Mani-
folds, defined, 122.
Inference and a Calculus, 10.
Infinity, Plane at, 497 ;
Points at, 506.
Inner Multiplication, 207 sqq,, 528.
Integrals, Volume, Surface, and Line, 562.
Intensity, 119 sqq,, 162 sqq, ;
and Secondary Properties, 15 ;
Complex, 120;
in Non-Euclidean Geometiy, 364, 366;
in Hyperbolic Geometry, of Points and
Planes, 415 sqq, ;
Intensity in Parabolic Geometry, 368, 498 ;
Locus of Zero, defined, 163 ;
Negative, 120 ;
Normal, defined, 200;
of Forces in Non-Euclidean Geometry,
399, 452 ;
Opposite, defined, 166.
Intensively Imaginary, or Beal, Elements, 166,
415.
Intercept, 167 sqq., 358 sqq.;
Antipodal, defined, 168 ;
Length of, defined, 359, 363 ;
Polar, defined, 358 ;
The, defined, 358.
Interpretation, Propositional, in Symbolic
Logic, 107.
Intersection of Manifolds, defined, 15.
Invariant Equations of Condition, 172.
Invariants of Groups of Systems, 300 sqq.,
531 sqq. ;
Conjugate, of Triple Groups, 310 sqq.;
Harmonic, of Dual Groups, 301, 533;
Null, of Dual Groups, 300 ;
Pole and Polar, of Triple Groups, 305,
584;
of Groups in Euclidean Space, 531 sqq.
Involution, Lines in, 280;
of Systems of Forces, 291 ;
Foci of, 291.
Jevons, 38, 39, 115.
Johnson, W. E., 28, 44, 48, 67, 73, 88, 116,
183.
Junghans, 574.
Killing, 370.
Kinematics in Euclidean Geometry, 536 sqq.,
551 sqq. ;
in Non-Euclidean Geometry, 456 sqq,
Klein, 127, 185, 278, 351, 353, 354, 869, 881,
456, 500, 501.
Koenigs, 278.
Kraft, F., 575.
Lachlan, 188.
Ladd, Miss Christine, 98, 116.
Lasker, E., 575.
Latent and Semi-Latent Regions, Types of, in
Three Dimensions, 317 sqq,;
Groups and Systems, Ilypes of, 326
sqq,;
Group, defined, 322 ;
Point, defined and discussed, 254 sqq,;
Regions, „ „ „ 248,256*99.;
582
INDEX.
Latent Regions, corresponding to roots oon-
joinUy, 316 ;
Boot, 254 ;
Repeated, 257 ;
System, defined and discussed, 822 sqq.
Law, Associative, and Combinatorial Multipli-
cation, 174 ;
and Matrices, 251 ;
and Symbolic Logic, 37 ;
and Universal Algebra, 25, 27 ;
Commutative, and Congruent Transforma-
tions, 464 ;
and Symbolic Logic, 37 ;
and Universal Algebra, 27 ;
Distributive, and Combinatorial Multipli-
cation, 174 ;
and Existential Expressions, 84 ;
and Symbolic Logic, 37 ;
and Universal Algebra, 26 ;
of Absorption, 37 ;
of Simplicity, 39 ;
Partial Suspension of, 88 ;
of Unity, 38 ;
Partial Suspension of, 88.
Laws of Thought, 110.
Letters, Greek, Roman, and Capital, Con-
ventions concerning, 86, 119, 177, 550 ;
Regional, 87 ;
Umbral, 86.
Leibnitz's Theorem, 273.
Lie, 369.
Limit Line, 495 ;
Surface, 447 sqq,, 486, 494.
Limited Field, defined, 61.
Limiting Equation, defined, 59.
Lindemann, 281, 369.
Line-Geometry, 278.
Lines, NuU, 278, 286;
Parallel, in Elliptic Space, 404 sqq. ;
in Hyperbolic Space, 436 sqq.;
Secant and Non-Secant, 436 ;
Spatial and Anti-Spatial, 418 ;
Straight, defined, 130.
Linear Complexes, 278 ;
and Quintuple Groups, 286 ;
Theorems concerning, 292 ;
Element, defined, 177 ;
compared to Mechanical Force, 273.
Lobatschewsky, 369, 486, 438, 487.
Locus, Containing, defined, 181 ;
Curvilinear, defined, 130 ;
defined, 128;
Flat, defined, 129 ;
of Zero Intensity, defined, 163 ;
Locus, Surface and Curve, defined, 13Q.
Logic, Application of Algebra to, 99 ;
Generalization of Formal, 106 ;
Symbolic, Algebra of , meniioDed,
Formal Laws of Algebra of, S3
Interpretations of Algebra of,
99, 107.
Lotze, 6, 116.
Love, 560.
MeCoU, 112, 116.
Magnitude, Extensive, defined, 176 tqq.
Bianifolds, 13 sqq. ;
Algebraic, 22, 26 sqq, ;
Orders of, 27, 171, 175 ;
Self-MultipUcative, 27 ;
Complete Algebraic System of, 27 ;
Derived, 175 ;
Extensive, and Elliptic Geometry, 399
tqq,;
and Hyperbolic Geometry, 452 sqq,;
mentioned, 31 ;
of Three Dimensions, 273 8qq. ;
Positional, investigated, 117 sqq. ;
mentioned, 30 ;
Spatial, defined, 349, 355 ;
Special, defined, 16 sqq.
Matrices, 248 sqq,, 816 sqq., 456 sqq., 500 sqq. ;
and Forces, 816 sqq, ;
Congruent, 457 sqq., 500, 536 ;
Denondnators and Numerators of, 249 ;
Null Spaces of, 252 ;
Nullity of, 253 ;
Skew, 248, 267 ;
symbolized, 280 ;
Spaces (or Regions) preserved by, 253 ;
Sums and Products of, 250 ;
Symmetrical, 248, 262 ;
Vacuity of, 261 ;
Vacuous Regions of, 262.
Maxwell, 573.
Metageometry, 369.
Metrics, Theory oi; 273, 347 sqq,
Meunier's Theorem, in Euclidean Space, 547;
in Non-Euclidean Space, 493.
Middle Factor, Extended Rule of the, 188 ;
Rule of the, 185;
for Inner Multiplication, 208.
MitcheU, Dr, 116.
Mixed Product, defined, 184.
Mobile Differential Operator, defined, 554.
Mdbius, 131.
Mode of a Property, 8, 13.
Moment of a System of Forces, 278.
•»
m
9
INDEX.
683
Monge» 479, 641.
Motion, AsBodated System with, 487.
MtlUer, 192.
Maltiplieation, Combinatorial, investigated, 171
sqqr,
FormulflB for, in Three Di-
mensions, 274 ;
Flux, 628;
in Symbolic Logic, interpreted, 88, 108;
Inner and Outer, 207 ;
in Eaolidean Space, 628 sqq.;
mentioned, 18;
Principles of, defined and difloossed, 26
sqq,;
Progressive and Regressive, 181 sqq,
Mcdtiplicative Combination, defined, 175.
Napier's Analogies and Non-Enolidean Geo-
metiy, 875, 425.
Negation, Primitive, 113.
Negative Intensity, 120 ;
System of Generators, 207.
Newcomb, 870.
Non-Eadidean Geometry, Historical Note
upon, 869 sgg.;
investigated, 347 <99-
Non-secant lines, 436.
Normal Intensity, defined, 200;
Points, defined, 199 sqq. ;
Regions, defined, 203 ;
and Points in Endidean (Geometry,
523 s^g.;
in Non-Endidean Geometry,
383 <9g., 426 <99.;
Systems of Points, defined, 200 ;
Rectangular, 524.
Nagatory Forms of Propodtions, 100.
Null Element and Sdf-Condemned Propodtions,
in Symbolic Logic, 35, 37 sqq.;
inteipxeted, 38, 109 ;
in Universal Algebra, 24 sqq,, 28;
Invariants of a Dual Group, 800 ;
Lines, Planes, and Points, 278;
Points, Latency of, 823 sqq. ;
Space of Matrix, 253 ;
Term, 24.
Nullity of Matrix, defined, 253.
Numbers, Alternate, 180.
One-dded Planes, 379.
Operation, General Definition of, 7 sqq.;
of Taking the Flux, 522, 527 ;
Vector, 516, 522.
Order and the Operation of Addition, 19 sqq.;
of Manifolds, 27, 171, 175 ;
of Tortuodty, 131.
Origin, defined, for Euclidean Geometry, 524;
for Hyperbolic Geometry, 414;
with Pure Vector Formul», 550.
Outer Multiplication, 207.
Oval Qnadrics, defined, 376;
Spheres, intersection of, 396.
Over-Strong Premises, 104.
Parabolic Definition of Distance, 353 sqq. ;
(Geometry, 355, 496 sqq. ;
as a Limiting Form, 367;
Group, 289, 296 ;
Semi-Latent, and Latent, 322 sqq. ;
Linear Transformation, 135 ;
Self-Supplementary Group, General Type
of, 296 ;
Subgroup of Triple Group, 296, 311 sqq.
Pamllel Lines in Elliptic Geometiy, 404 sqq.;
in Hyperbolic Geometry, 436 sqq.;
Planes in Hyperbolic Geometiy, 439;
Regions in Elliptic Geometry, 397 sqq.
Parallelism, Angle of, 438 ;
Right and Ldt, 405.
Parallelogram of Forces, 273 ;
in ElUptio Space, 410.
Parameters of a Congruent Transformation,
460, 471 ;
of a System of Forces, 401, 454.
Partition of a Manifold, 15.
Pascal^s Theorem, 231, 237.
Peano, G., 575.
Peirce, B., 172 ;
0. S., 3, 10, 37, 42, 115.
Perpendiculars in Non-Eudidean Geometry,
383 sqq.j 426 sqq.
Perspective, 139 sqq,
Pfafl's Equation, 573.
Planar Elements, defined, 177 ;
Intendties of; in Non-Eudidean Geo-
metry, 366, 415.
Planes, Angles between, in Non-Eudidean
Geometry, 365, 382, 417 sqq. ;
Central, of Spheres, 442 ;
Defined, 130;
Diametral, of Groups, 532 ;
Paralld, in Hyperbolic Space, 439 ;
One-sided, 379 sqq,
Pliicker, 213, 278.
Polar Form of Elliptic Geometry, defined,
355;
Intercept, defined, 858 ;
584
INDEX.
Polar Invamnt, $ee Invariant ;
Bedprooally, defined, 145 ;
Regions, Absolute, 367, 384, 420 ;
Self-, defined, 145.
Pole Inyariant, 8ee Invariant.
Pole and Polar, 145 ;
in Hyperbolic Geometry, 420.
Poincar6, 369.
Point and Vector Factors, 614 ;
Inside a Triangle, defined, 375 ;
Latent, 254, 257, 317.
Points, Normal, 199 8qq. ;
Normal or qoadrantal, in Non-Enclidean
Geometry, 883, 414, 426.
Positional Manifolds, defined, 30;
investigated, 117 sqq.
Positive System of Generators, 207.
Premises, Over-Strong, 104.
Primitive Predication and Negation, 112 sqq.
Principal Systems of a Daal Ghronp, defined,
532;
Triangles, 373 8qq.,d80 sqq.;
Vector of a System of Forces, 518.
Products, Order of, defined, 175 ;
Pore and mixed, 184.
Progressive Multiplication, 181.
Projection, 224 sqq,
Propositional Interpretation of Symbolic Logic,
107 sqq.
Propositions, Equivalence of, defined, 108 ;
Nugatory Forms of, 100 ;
Bedprooal, 38 ;
Self-condemned, defined. 111 ;
Simple, 107 ;
Symbolic Forms of, 99 aqq.^ Ill sqq.
Pure Products, defined, 184.
Qnadrantal Points, 383.
Quadrics, 144 sqq. ;
Absolute, 355 ;
and Inner Multiplication, 210 ;
Associated with Triple Groups, see
Group;
Conical, defined, 150 ;
investigated, 155 ;
Closed, 153, 355, 376 ;
Line-Equation of, generalized, 213 ;
Oval, defined, 376;
Self-normal, defined, 199 ;
Extended Definition of, 201 ;
Transformation of. Direct and Skew,
338 sqq.
Quaternions, mentioned, 32, 115, 131, 554, 573.
Quotient, name for Matrix, 249.
Batio, Anhannonio, defined, 132 ;
of Systems of f*oroes, 291.
Beciprocal Groups, 285 ;
Propositions, 38 ;
Systems of Forces, 281, 303;
and Work, 469, 477, 638.
Reciprocity between Addition and Mnltiplieatiaa
in Symbolic Logic, 37.
Rectangular Normal Systems in EneUdeu
Space, 524 ;
Rectilinear Figures in H7peii»&
Space, 433.
Reference Figures, 138.
Regional Element, in Calcolns of Bxteosiao,
177;
Letters, in Symbolic Zjogic, 87.
Regions and Symbolic Logic, 38 ;
Incident, 42;
Complete, defined, 123 ;
Containing, 126;
Co-ordinate, 12G;
Generating, 147 ;
and Non-Euclidean Oeometiy, 897,
451;
Latent and Semi-Latent, 248, 256 sqqr,
corresponding to BootB coujoinUj,
316;
Types of, in Three DimensioD^
317 sqq. ;
Mutually Normal, 203 ;
Non-vertical, 159;
Null, of Matrices, 252 ;
Parallel, 397, 451 ;
Preserved by Matrices, 253 ;
Semi-Latent, see Regions, Latent ;
Supplementary, 126;
Symbolism for, in Calcalns of Exten-
sion, 177 ;
Vacuous, of Matrices, 262.
Regressive Multiplication, 181 sqq.
Relation, General Definition of, 8.
Resultants, defined, 51 sqq.\
Existential, 90 sqq. ;
of Subenmptions, 59.
Riemann, 13, 17, 368, 369.
Rigid Body, Motion of, in Non-Eudidesn
Geometry, 487.
Rotation in Euclidean Geometry, 502 ;
in Non-Euclidean Geometry, 460, 47t
Russell, B. A. W., 369, 370.
Salmon, 151.
Scalar, 552.
INDEX.
585
Sdhemea, Algebnio, 92, 36, 85 ;
Subatitativd, 8 sqq.^ 14.
Sohlegel, Y., 576.
Sohrdder, 87, 42, 51, 62, 66, 67, 74, 82, 115, 116.
Scott, B. F., 180.
Secant Lines, 436.
Secondary Properties of Elements, 14, 120 sqq.;
Triangles, 878.
Self-condemned Propositions, defined. 111.
Self-Conjugate Tetrahedrons and Conjngate
Sets of Systems, 808.
Self-normal Elements, 199, 204 ;
Qnadric, 199. 201, 204 ;
Sphere, 524.
Semi-Latent, see Latent.
Shadows, 87 sqq, ;
Weak Forms of, 84.
Sign, Standard, 863, 416.
Signs and Symbolism, 8 sqq.
Simple Extensive Magnitude, defined, 177.
Simplicity, Law of, 39 ;
Partial Suspension of, 88.
Skew Transformation of a Qnadric, 888, 842
sqq.
Space and Symbolic Logic, 80 sgg., 88 sqq, ;
Constant, 868;
defined, 854 ;
Descriptive Properties of, 119 sqq, ;
Division of, 855, 879;
Elliptic, Hyperbolic, and Parabolic, 855
sqq,;
Enclidean, 505 sqq,;
Flatness of, 451 ;
Ideal, 414 ;
Non-Euclidean, Formulaa for, 856 sqq,^
862 sqq,;
Null, of a Matrix, 252;
Preserved by a Matrix, 258 ;
Vacuous, of a Matrix, 262.
Spaces as Spherical Loci, 450 sqq.
Spatial Elements, 414 sqq.;
Interpretation of Universal Algebra, 81 ;
Manifolds, defined, 854 sqq.
Species of Algebras, 27.
Sphere, defined, 876 ;
investigated, 891 sqq.^ 441 sqq. ;
Self-normal, 524.
Spherical Geometry, 855.
Stiiokel, 869, 870.
Standard Form of Planes, 415 ;
Sign, see Sign.
Stationary Differential Operator, 558.
Staudt, von, 215, 817.
Staudtian, Formnlie for, 874.
W.
Steps, 25, 507.
Stereometrical Triangles, 882, 425.
Stokes* Theorem, 562.
Stout, 8, 4.
Sturm, Dr Bndolf, 278.
Subgroup, Parabolic, of Triple Qronp, 296, 811.
Submanifold, defined, 15.
Subplane, defined, 180.
Subregions, defined and investigated, 128, 125 ;
Distances of, 865 ;
Parallel, 397, 451.
Substitutive Schemes, 8 ;
Signs, 8 sqq.
Subsnmptions, defined and investigated, 42
sqq.f 59.
Subtraction and Symbolic Logic, 80 ;
and Universal Algebra, 22 sqq., 28.
Supplementary Terms, interpreted, 89, 110.
Supplements and Reciprocal Polars, 201 ;
defined, 181 sqq.;
Different kinds of, 202 ;
Extended Definition of, 201 ;
in EncUdean Space, 528 sqq.;
in Symbolic Logic, 86 sqq.;
investigated, 199 sqq.
Surface Locus, defined, 180.
Surfaces in Euclidean Space, 589 sqq. ;
in Non-Endidean Space, 488 sqq.;
of Equal Distance from Planes, 891, 442;
from Subregions, 897, 451.
SyllogiEon, 101 ;
Qeneralization of, 105 ;
Symbolic Equivalents of, 108.
Sylvester, 154, 253, 254.
Symbolic Logic, Formal Laws of Algebra of,
35 sqq. ;
interpreted, 38, 99 sqq*^ 107 sqq. ;
mentioned, 22, 29.
Synthesis, General Definition of, 8.
Systems of Forces, 278 sqq. ;
and Quadrics, 294 sqq., 298, 888, 535;
associated with Congruent Transform-
ation, 466, 478, 476, 536;
Axes of, 401, 454 ;
Groups of, see Groups ;
in Euclidean Space, 518 sqq.;
in Non-EucUdean Space, 401 sqq., 454
in Notation of Pure Vector Formulie,
551;
Latent, defined, 322 ;
Non-Axal, 404, 455 ;
Parameters of, 401, 454 ;
Bedprocal, 281 ;
38
586
INDEX.
Systems, Beoiprocal, and Work, 469, 477, 538;
Vector, 406 «gg., 464;
and Vector Transformations, 478 ;
Unit, 407.
Tait, 554, 573.
Taylor, H. M., 215.
Terms, defined, 20 ;
and Points, distinguished, 360 ;
Constituent, defined, 47 ;
Congruence of, defined, 122 ;
defining Triangle, 872 ;
Equivalence of, interpreted, 38 ;
NuU, 24 ;
Supplementary, interpreted, 39.
Thought, Laws of, 110.
Tortuosity, Order of, 131.
Transformation, Congruent, 456, 500, 537 ;
and Characteristic Lines,
470;
and Work, 469, 477, 687 ;
Associated Systems of
Forces of, 466, 473, 476,
536;
Axes of, 458, 471 ;
Parameters of, 460, 471 ;
EUiptio, Hyperbolic, Parabolic, 134;
Linear, 133, 227, 248, 316 ;
of a Quadric, Direct and Skew, 338,
466 sqq. ;
Vector, 472 sqq.
Translation in Eudidean Space, 502 ;
in Non-Euclidean Space, 460, 471.
Triangles, 371 sqq., 378, 422 sqq.\
defined by Terma, 372;
Principal, 373, 380 ;
Set of, 373 ;
Secondary, 373 ;
Spatial and Semi-Spatial, 423 ;
Stereometrical, 425, 482.
Trigonometiy, Spherical, 375, 383, 426.
Umbral Letters, 86 sqq.
Uninterpretable ExpressioEis io a CsLetdoA, Vk
Units, Independent, defined, 122.
Unity, Law of, 38 ;
Partial Suspension of, 88.
Universal Algebra, 11, 18 ;
Propositions and Equationa, 105.
Universe and Primitive Predioation, 113 ;
defined, 36 ;
interpreted, 39, 109 ;
Properties of, 37 ;
of Discourse, 100.
Unlimiting Equations, 59 sqq.
Vacuity of a Matrix, 261.
Vacuous Region of a Matrix, 262.
Vector in EUiptic Space, viz.
Vector System of Forces, 406 sqq., 454, 473 ;
Associated with Molioii, 473 ;
Right and Left, 406 gqq. ;
Unit, defined, 407 ;
Transformations, 472.
Vector in Euclidean Space, viz.
Vector Analysis, 573 ;
Areas, 509 sqq. ;
Differentiation, 560 ;
Factor, 514 ;
Flow of a, 572 ;
Formuls, Pure, 548 sqq,;
Moment, 518, 551 ;
Operation of Taking the, 616 sqq., 532 ;
Potential, 565 ;
Principal, of System of Foroee, 518 ;
Volumes, 513 sqq.
Velocity in Non-Euclidean Spaoe, 482.
Venn, 115, 116.
Veronese, 139, 147, 152, 161, 370.
Vortex Motion, 562.
Weak forms of Shadow Letters, 84.
Weakened Condnsion, 104.
Webb, 560.
Work in Eudidean Spaoe, 637 ;
in Non-Euclidean Space, 468, 477.
CAMBBIDOB: PBINTVD by J. and O. p. clay, at the UNITERSITY P1UC88.
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